CHARACTERIZATION OF SEDIMENT AND GRAIN SIZE DISTRIBUTIONS

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CHAPTER 2 CHARACTERIZATION OF SEDIMENT AND GRAIN
SIZE DISTRIBUTIONS
Sediment diameter is denoted as D the parameter
has dimension L. Since sediment particles are
rarely precisely spherical, the notion of
diameter requires elaboration. For
sufficiently coarse particles, the diameter D
is often defined to be the dimension of the
smallest square mesh opening through which the
particle will pass. For finer particles,
diameter D often denotes the diameter of the
equivalent sphere with the same fall velocity vs
L/T as the actual particle. For reasons that
will become apparent below, grain size is often
specified in terms of a base-2 logarithmic scale
(phi scale or psi scale). These are defined as
follows where D is given in mm,
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SAMPLE EVALUATIONS OF ? AND ?
D (mm) ? ?
4 2 -2
2 1 -1
1 0 0
0.5 -1 1
0.25 -2 2
0.125 -3 3
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SEDIMENT SIZE RANGES
Type D (mm) ? ? Notes
Clay lt 0.002 lt -9 gt 9 Usually cohesive
Silt 0.002 0.0625 -9 -4 4 9 Cohesive non-cohesive
Sand 0.0625 2 -4 1 -1 4 Non-cohesive
Gravel 2 64 1 6 -6 -1
Cobbles 64 256 6 8 -8 -6
Boulders gt 256 gt 8 lt -8
Mineral clays such as smectite, montmorillonite
and bentonite are cohesive, i.e. characterized by
electrochemical forces that cause particles to
stick together. Even silt-sized particles that
are do not consist of mineral clay often display
some cohesivity due to the formation of a
biofilm.
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SEDIMENT GRAIN SIZE DISTRIBUTIONS
The grain size distribution is characterized in
terms of N1 sizes Db,i such that ff,i denotes
the mass fraction in the sample that is finer
than size Db,i. In the example below N 7.
i Db,i mm ff,i
1 0.03125 0.020
2 0.0625 0.032
3 0.125 0.100
4 0.25 0.420
5 0.5 0.834
6 1 0.970
7 2 0.990
8 4 1.000
Note the use of a logarithmic scale for grain
size.
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SEDIMENT GRAIN SIZE DISTRIBUTIONS contd.
In the grain size distribution of the last slide,
the finest size (0.03125 mm) was such that 2
percent, not 0 percent was finer. If the finest
size does not correspond to 0 percent content, or
the coarsest size to 100 percent content, it is
often useful to use linear extrapolation on the
psi scale to determine the missing values.
i Db,i mm ff,i
1 0.0098 0
2 0.03125 0.020
3 0.0625 0.032
4 0.125 0.100
5 0.25 0.420
6 0.5 0.834
7 1 0.970
8 2 0.990
9 4 1.000
i Db,i mm ff,i
1 0.03125 0.020
2 0.0625 0.032
3 0.125 0.100
4 0.25 0.420
5 0.5 0.834
6 1 0.970
7 2 0.990
8 4 1.000
Note that the addition of the extra point has
increased N from 7 to 8 (there are N1 points).

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SEDIMENT GRAIN SIZE DISTRIBUTIONS contd.
The grain size distribution after extrapolation
is shown below.
i Db,i mm ff,i
1 0.0098 0
2 0.03125 0.020
3 0.0625 0.032
4 0.125 0.100
5 0.25 0.420
6 0.5 0.834
7 1 0.970
8 2 0.990
9 4 1.000
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CHARACTERISTIC SIZES BASED ON PERCENT FINER
Dx is size such that x percent of the sample is
finer than Dx Examples D50 median size D90
roughness height
To find Dx (e.g. D50) find i such that
Then interpolate for ?x
and back-calculate Dx in mm
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STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION
N1 bounds defines N grain size ranges. The ith
grain size range is defined by (Db,i, Db,i1) and
(ff,i, ff,i1)
?i (Di) characteristic size of ith grain size
range fi fraction of sample in ith grain size
range
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STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION
contd.
mean grain size on psi scale ? standard
deviation on psi scale
Dg geometric mean size ?g geometric standard
deviation ( ? 1) Sediment is well sorted if ?g lt
1.6
Dg 0.273 mm, ?g 2.17
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GRAIN SIZE DISTRIBUTION CALCULATOR
A key feature of this e-book is the library of
Excel spreadsheet workbooks that go with it.
These workbooks allow for implementation of the
formulations given in the PowerPoint
lectures. Some of these workbooks allow for
calculations to be performed directly on the
worksheets of the workbook. Others use one or
more worksheets as GUIs (Graphical User
Interfaces), where the click of a button executes
a code in VBA (Visual Basic for Applications)
that is imbedded in the workbook. The first such
workbook of this e-book is RTe-bookGSDCalculator.x
ls. It computes the statistics of a grain size
distribution input by the user, including Dg, ?g,
and Dx where x is a specified number between 0
and 100 (e.g. the median size D50 for x 50).
It uses code in VBA (macros) to perform the
calculations. You will not be able to use macros
if the security level in Excel is set to High.
To set the security level to a value that allows
you to use macros, first open Excel. Then click
Tools, Macro, Security and then in
Security Level check Medium. This will
allow you to use macros.
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GRAIN SIZE DISTRIBUTION CALCULATOR contd.
When you open the workbook RTe-bookGSDCalculator.x
ls, click Enable Macros. The GUI is contained
in the worksheet Calculator. Now to access the
code, from any worksheet in the workbook click
Tools, Macro, Visual Basic Editor. In the
Project window to the left you will see the
line VBA Project (FDe-bookGSDCalculator.xls).
Underneath this you will see Module1.
Double-click on Module1 to see the code in the
Code window to the right. These actions allow
you to see the code, but not necessarily to
understand it. In order to understand this
course, you need to learn how to program in VBA.
Please work through the tutorial contained in the
workbook RTe-bookIntroVBA.xls. It is not very
difficult! All the input are specified in the
worksheet Calculator. First input the number
of pairs npp of grain sizes and percents finer
(npp N1 in the notation of the previous
slides) and click the appropriate button to set
up a table for inputting each pair (grain size in
mm, percent finer) in order of ascending size.
Once this data is input, click the appropriate
button to compute Dg and ?g. To calculate any
size Dx where x denotes the percent finer, input
x into the indicated box and click the
appropriate button. To calculate Dx for a
different value of x, just put in the new value
and click the button again.
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GRAIN SIZE DISTRIBUTION CALCULATOR contd.
This is what the GUI in worksheet Calculator
looks like.
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GRAIN SIZE DISTRIBUTION CALCULATOR contd.
If the finest size in the grain size distribution
you input does not correspond to 0 percent finer,
or if the coarsest size does not correspond to
100 percent finer, the code will extrapolate for
these missing sizes and modify the grain size
distribution accordingly. The units of the code
are Subs (subroutines). An example is given
below.
Sub fraction(xpf, xp) 'computes fractions
from finer Dim jj As Integer
For jj 1 To np xp(jj) (xpf(jj) -
xpf(jj 1)) / 100 Next jj End Sub
In this Sub, xpf denotes a dummy array containing
the percents finer, and xp denotes a dummy array
containing the fractions in each grain size
range. The Sub computes the fractions from the
percents finer. Suppose in another Sub you know
the percents finer Ff(i), I 1..npp and wish to
compute the fraction in each grain size range
F(i), i 1..np (where np npp 1). The
calculation is performed by the
statement fraction Ff, f
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WHY CHARACTERIZE GRAIN SIZE DISTRIBUTIONS IN
TERMS OF A LOGARITHMIC GRAIN SIZE?
Consider a sediment sample that is half sand,
half gravel (here loosely interpreted as material
coarser than 2 mm), ranging uniformly from 0.0625
mm to 64 mm. Plotted with a logarithmic grain
size scale, the sample is correctly seen to be
half sand, half gravel. Plotted using a linear
grain size scale, all the information about the
sand half of the sample is squeezed into a tiny
zone on the left-hand side of the diagram.
Logarithmic scale for grain size
Linear scale for grain size
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UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS
The fractions fi(?i) represent a discretized
version of the continuous function f(?), f
denoting the mass fraction of a sample that is
finer than size ?. The probability density pf of
size ? is thus given as p df/d?.
The example to the left corresponds to a Gaussian
(normal) distribution with -1 (Dg 0.5
mm) and ? 0.8 (?g 1.74)
The grain size distribution is called unimodel
because the function p(?) has a single mode, or
peak.
The following approximations are valid for a
Gaussian distribution
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UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS
contd.
A sand-bed river has a characteristic size of bed
surface sediment (D50 or Dg) that is in the sand
range. A gravel-bed river has a characteristic
bed size that is in the range of gravel or
coarser material. The grain size distributions
of most sand-bed streams are unimodal, and can
often be approximated with a Gaussian
function. Many gravel-bed river, however, show
bimodal grain size distributions, as shown to the
upper right. Such streams show a sand mode and a
gravel mode, often with a paucity of sediment in
the pea-gravel size (2 8 mm).
Plateau
Gravel mode
Sand mode
A bimodal (multimodal) distribution can be
recognized in a plot of f versus ? in terms of a
plateau (multiple plateaus) where f does not
increase strongly with ?.
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UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS
contd.
The grain size distributions to the left are all
from 177 samples from various river reaches in
Alberta, Canada (Shaw and Kellerhals, 1982). The
samples from sand-bed reaches are all unimodal.
The great majority of the samples from gravel-bed
reaches show varying degrees of
bimodality. Note geographers often reverse the
direction of the grain size scale, as seen to the
left.
Figure adapted from Shaw and Kellerhals (1982)
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GRAVEL-SAND TRANSITIONS
As rivers flow from mountain reaches to plains
reaches, sediment tends to deposit out, creating
an upward concave long profile of the bed and a
pattern of downstream fining of bed sediment.
Both these patterns are evident in the plots to
the right for the Kinu River, Japan (Yatsu,
1955). It is common (but by no means universal)
for fluvial sediments to be bimodal, with sand
and gravel modes and a relative paucity in the
range of pea gravel. In such cases a relatively
sharp transition from a gravel-bed stream to a
sand-bed stream is often found, often with a
concomitant break in slope (Sambrook Smith and
Ferguson, 1995, Parker and Cui, 1998). Both
these features are evident for the Kinu River.
Long profiles of bed elevation, bed slope and
median grain size for the Kinu River, Japan.
Adapted from Yatsu (1955)
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VERTICAL SORTING OF SEDIMENT
Gravel-bed rivers such as the River Wharfe often
display a coarse surface armor or pavement.
Sand-bed streams with dunes such as the one
modeled experimentally below often place their
coarsest sediment in a layer corresponding to the
base of the dunes.
River Wharfe, U.K. Image courtesy D. Powell.
Sediment sorting in a laboratory flume. Image
courtesy A. Blom.
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SEDIMENT DENSITY
Definitions ? density of water ?s material
density of sediment s ?s/? specific gravity
of sediment R (?s/?) 1 submerged specific
gravity of sediment
The default sediment density is that of quartz,
i.e. 2.65 grams/cm3. This corresponds to the
values s 2.65 and R 1.65. Two other common
natural rock types are basalt (s 2.7 2.9) and
limestone (s 2.6 2.8). Volcanic sediment
often have vugs (large pores), which reduce their
effective specific gravity to lower values (e.g.
2.0 in the case of pumice the value can be less
than 1.) Rocks containing heavy minerals such as
magnetite can have specific gravities of 3
5. It is common to use lightweight model
sediments in the laboratory. Examples include
crushed walnut shells (s 1.4), crushed coal (s
1.3 1.5) and plastic particles (s 1 2).
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SEDIMENT FALL VELOCITY IN STILL WATER
Assume a spherical particle with diameter D and
fall velocity vs. The downstream impelling force
of gravity Fg is
means cD is a function of Revp see any good
fluid mechanics text
The resistive drag force is
where ? is the kinematic viscosity of the water
and cD is specified by the empirical drag curve
for spheres.
Condition for equilibrium
where
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SEDIMENT FALL VELOCITY IN STILL WATER contd.
Untangle the relation
where
and
where
Again this means a functional relationship
Reduce to Rf Rf(Rep) Relation of Dietrich
(1982)
The original relation also includes a correction
for shape.
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SOME SAMPLE CALCULATIONS OF SEDIMENT FALL
VELOCITY (Dietrich Relation)

• g 9.81 m s-2
• R 1.65 (quartz)
• 1.00x10-6 m2 s-1 (water at 20 deg Celsius)
• ? 1000 kg m-3 (water)

The calculations to the left were performed with
RTe-bookFallVel.xls. Have a look at it. This
Excel workbook implements the Dietrich (1982)
fall velocity relation. It does not use macros
to perform the calculation. In a later chapter of
this e-book, this workbook is used to compute
fall velocities in the implementation of
calculations of suspended sediment concentration
profiles.
D, mm vs, cm/s
0.0625 0.330
0.125 1.08
0.25 3.04
0.5 7.40
1 15.5
2 28.3
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USE OF THE WORKBOOK FDe-bookFallVel.xls
A view of the interface in RTe-bookFallVel.xls is
given below. Since VBA is not used, it is not
necessary to click a button to perform the
calculations. Just fill in the input cells, and
the answer will appear in the output cell.
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MODES OF TRANSPORT OF SEDIMENT
Bed material load is that part of the sediment
load that exchanges with the bed (and thus
contributes to morphodynamics). Wash load is
transported through without exchange with the
bed. In rivers, material finer than 0.0625 mm
(silt and clay) is often approximated as wash
load. Bed material load is further subdivided
into bedload and suspended load. Bedload sliding
, rolling or saltating in ballistic trajectory
just above bed. role of turbulence is
indirect.   Suspended load feels direct
dispersive effect of eddies. may be wafted high
into the water column.
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REFERENCES FOR CHAPTER 2
Dietrich, E. W., 1982, Settling velocity of
natural particles, Water Resources Research, 18
(6), 1626-1982. Parker. G., and Y. Cui, 1998, The
arrested gravel front stable gravel-sand
transitions in rivers. Part 1 Simplified
analytical solution, Journal of Hydraulic
Research, 36(1) 75-100. Sambrook Smith, G. H.
and R. Ferguson, 1995, The gravel-sand transition
along river channels, Journal of Sedimentary
Research, A65(2) 423-430. Shaw, J. and R.
Kellerhals, 1982, The Composition of Recent
Alluvial Gravels in Alberta River Beds, Bulletin
41, Alberta Research Council, Edmonton, Alberta,
Canada. Yatsu, E., 1955, On the longitudinal
profile of the graded river, Transactions,
American Geophysical Union, 36 655-663.