1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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,

D2 2

n(D)
    og2 (D) 
n(2)
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE EVALUATIONS OF  AND 

D2 2
n(D)
    og2 (D) 
n(2)

D (mm)


4
2
-2
2
1
-1
1
0
0
0.5
-1
1
0.25
-2
2
0.125
-3
3
2
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SEDIMENT SIZE RANGES
Type
D (mm)


Notes
Clay
< 0.002
< -9
>9
Usually cohesive
Silt
0.002 ~ 0.0625 -9 ~ -4
4~9
Cohesive ~ noncohesive
Sand
0.0625 ~ 2
-4 ~ 1
-1 ~ 4
Non-cohesive
Gravel
2 ~ 64
1~6
-6 ~ -1
“
Cobbles
64 ~ 256
6~8
-8 ~ -6
“
>8
< -8
“
Boulders > 256
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.
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SEDIMENT GRAIN SIZE DISTRIBUTIONS
The grain size distribution is
characterized in terms of N+1
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.
Sample Grain Size Distribution
100
90
Percent Finer
80
70
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
Grain Size mm
6
1
0.970
Note the use of a logarithmic
scale for grain size.
7
2
0.990
8
4
1.000
60
50
100 x ff,4 = 42
40
30
Db,4 = 0.25 mm
20
10
0
0.01
0.1
1
10
4
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
og2 (Db,2 )  og2 (Db,3 )
0  ff ,3 
b,1  og2 (Db,3 ) 
ff ,2  ff ,3
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
b,i  n(Db,i ) n(2)  og2 (Db,i )
Note that the addition
of the extra point has
increased N from 7 to
8 (there are N+1
points).
b ,1
Db,1  2
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
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SEDIMENT GRAIN SIZE DISTRIBUTIONS contd.
The grain size distribution after extrapolation is shown below.
Sample Grain Size Distribution (with Extrapolation)
i
Db,i mm
ff,i
90
1
0.0098
0
80
2
0.03125
0.020
70
3
0.0625
0.032
60
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
Percent Finer
100
50
100 x ff,5 = 42
40
30
Db,5 = 0.25 mm
20
10
0
0.001
0.01
0.1
Grain Size mm
1
10
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHARACTERISTIC SIZES BASED ON PERCENT FINER
Sample Grain Size Distribution (with Extrapolation)
100
D90 = 0.700 mm
90
Dx is size such that x percent of
the sample is finer than Dx
Examples:
D50 = median size
D90 ~ roughness height
Percent Finer
80
To find Dx (e.g. D50) find i such that
70
60
50
ff ,i 
D50 = 0.286 mm
40
Then interpolate for x
30
20
10
0
0.001
x
 ff ,i1
100
0.01
0.1
Grain Size mm
1
10
 b,i1   b,i  x

 x   b,i 
 ff ,i 

ff ,i1  ff ,i  100

and back-calculate Dx in mm
D x  2 x
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION
Sample Grain Size Distribution (with Extrapolation)
100
90
1
 i   b,i   b,i1 
2
1/ 2
Di  Db,iDb,i1 
Percent Finer
80
70
60
f5 = ff,6 - ff,5 = 0.414
50
40
D5 = (Db,5 Db,6)1/2
= 0.354 mm
30
20
fi  ff ,i1  ff ,i
i (Di) = characteristic size of ith
grain size range
10
0
0.001
N+1 bounds defines N grain size
ranges. The ith grain size range
is defined by (Db,i, Db,i+1)
and (ff,i, ff,i+1)
0.01
0.1
Grain Size mm
1
10
fi = fraction of sample in ith grain
size range
8
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION contd.
Sample Grain Size Distribution (with Extrapolation)
100
 = standard deviation on psi scale
N
    i fi
90
80
Percent Finer
 = mean grain size on psi scale
70
i1
60
N
 2    i    fi
50
i1
40
Dg  2 
30
20
 g  2
10
0
0.001
2
0.01
0.1
1
10
Grain Size mm
Dg = geometric mean size
Dg = 0.273 mm, g = 2.17
g = geometric standard deviation (  1)
Sediment is well sorted if g < 1.6
9
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 GUI’s
(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.xls. 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.
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GRAIN SIZE DISTRIBUTION CALCULATOR contd.
When you open the workbook RTe-bookGSDCalculator.xls, 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 (FDebookGSDCalculator.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 = N+1 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 11
value and click the button again.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
GRAIN SIZE DISTRIBUTION CALCULATOR contd.
This is what the GUI in worksheet “Calculator” looks like.
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 “Sub”s (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
13
fraction Ff, f
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
Grain Size Distribution: Half Sand, Half Gravel
0.0625 ~ 64 mm, linear scale
100
100
90
90
80
80
70
sand
Percent Finer
Percent Finer
Grain Size Distribution: Half Sand, Half Gravel
0.0625 mm ~ 64 mm, Logarithmic Scale
gravel
60
50
40
30
70
gravel
60
50
40
30
20
20
10
10
0
0.01
sand
0
0.1
1
10
D mm
100
0
10
20
30
40
50
60
D mm
14
Logarithmic scale for grain size
Linear scale for grain size
70
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
1
The example to the left
corresponds to a Gaussian
(normal) distribution with  = -1
(Dg = 0.5 mm) and  = 0.8 (g =
1.74):
p
 1    
1
exp  

2

2 



2



0.9
f()
0.8
0.7
0.6
0.5
0.4
0.3
p()
0.2
0.1
0
-4
-3
-2
-1
0
1
2

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:
D84
15
Dg  D84D16 , g 
D16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS contd.
1
A sand-bed river has a characteristic
size of bed surface sediment (D50 or Dg) 0.9
0.8
that is in the sand range.
f()
0.7
Plateau
0.6
A gravel-bed river has a characteristic
0.5
bed size that is in the range of gravel or
0.4
coarser material.
0.3
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).
Gravel mode
Sand mode
p()
0.2
0.1
0
-4
-2
0
2
4
6
8

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 .
16
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 sandbed 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)
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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).
Long profiles of bed elevation,
bed slope and median grain size
for the Kinu River, Japan.
Adapted from 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 sandbed 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.
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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).
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SEDIMENT FALL VELOCITY IN STILL WATER
Assume a spherical particle with diameter D and fall
velocity vs.
FD
Fg
The downstream impelling force of gravity Fg is:
4
D
Fg  Rg 
3
2
3
means cD is a function of Revp:
see any good fluid mechanics text
The resistive drag force is
2
v sD
1
D 2
FD  c D   v s , c D  c D Rev p  , Rev p 
2

2
where  is the kinematic viscosity of the water and cD is
specified by the empirical drag curve for spheres.
Condition for equilibrium:
Fg  FD
4
Rf  [
]1/ 2
3c D (Rev p )
where
vs
Rf 
RgD
21
Rf  [
4
]1/ 2
3c D (Re p )
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SEDIMENT FALL VELOCITY IN STILL WATER contd.
Untangle the relation:
v sD
vs
where R f 
and Re v p 

RgD
FD
4
Rf  [
]1/ 2
3c D (Rev p )
Fg
RgD D
v sD
vs
Rev p 

 R f Rep


RgD
Reduce to Rf = Rf(Rep)
Again this means a functional
relationship
Relation of Dietrich (1982):
R f  exp { b1  b2 n (Re p )  b3 [n (Re p )]
2
 b 4 [n (Re p )] 3  b5 [n (Re p )] 4}
The original relation also includes a correction for shape.
where Rep 
RgD D

b1
2.891394
b2
0.95296
b3
0.056835
b4
0.002892
b5
0.000245
22
Rf  [
4
]1/ 2
3c D (Re p )
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
D, mm
vs, cm/s
Have a look at it.
0.0625
0.330
0.125
1.08
0.25
3.04
0.5
7.40
1
15.5
2
28.3
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.
23
Rf  [
4
]1/ 2
3c D (Re p )
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
24
Rf  [
4
]1/ 2
3c D (Re p )
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
25
Rf  [
4
]1/ 2
3c D (Re p )
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
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