Middleton G.V. (Ed.) Encyclopedia of Sediments and Sedimentary Rocks
Подождите немного. Документ загружается.
GRAIN FLOW
335
Although graded bedding
is
generally used
to
describe
upward changes
in
grain size,
it is
permissable
to
extend
the
concept
to
compositional grading,
for
example
in
pyroclastic
deposits
(Cas and
Wright,
1987, p. 194).
White etal. (2001)
describe reverse grading
in
marine pyroclastic deposits, formed
because larger pumice takes longer
to
become saturated with
water
(and
sink) than smaller particles.
The term "grading"
is
generally used
to
describe vertical
changes
in
average grain size. However,
it is
acceptable
to
refer
to lateral grading
if
the grain size changes systematically along
or across
the
transport path.
Richard
N.
Hiscott
Bibliography
Allen, J.R.L., 1984. Sedimentary Structures: Their Character and
Physi-
cal Basis, Volume
It.
Amsterdam: Elsevier.
Bailey, E.B., 1930. New light
on
sedimentation and teetonics. Geological
Magazine,
67:
77-92.
Bagnold,
R.A.,
t956.
The
flow
of
cohesionless grains
in
fluids.
Philosophical Transactions of the Royal Society
of
London
(A), 249:
235-97.
Cas,
R.A.F.,
and
Wright,
J.V., 1987.
Volcanic
Successions: Modern
and Ancient:
A
Geological Approach
to
Processes, Products
and
Successions. London: Unwin-Hyman.
Hiseott,
R.N., 1994.
Loss
of
capacity,
not
competence,
as the
fundamental process governing deposition from turbidity currents.
Journatof Sedimentary Research,
A64:
209-214.
Kneller,
B.,
1995. Beyond
the
turbidite paradigm: physical models
for
deposition
of
turbidites
and
their implications
for
reservoir predic-
tion,
tn
Hartley,
A.J., and
Prosser,
D.J.
(eds.). Characterization of
Deep Marine Clastic Systems. Oxford: Geological Society (London),
Special Publication,
94, pp.
31-50.
Labaume,
P.,
Mutti,
E., and
Seguret,
M., 1985.
Megaturbidites:
a
depositional model from
the
eocene
of the
SW-Pyrenean foreland
basin, Spain. Geo-MarineLetters,
7:
91-lOL
Middleton,
G.V.,
1967. Experiments
on
density
and
turbidity currents:
in. deposition
of
sediment. Canadian Journal of Earth Sciences,
4:
475-505.
Niehols,
G., 1999.
Sedimentology
and
Stratigraphy. Oxford: Blackwell
Scientific.
Walker,
R.G., 1965. The
origin
and
significance
of the
internal
sedimentary structures
of
turbidites. Yorkshire Geological Society,
Proceedings,
35: 1-32.
Walker,
R.G., 1975.
Generalized facies model
for
resedimented
conglomerates
of
turbidite association. Geological Society
of
America,
Bulletin,
86: Til-lAi.
Weaver, P.P.E.,
and
Rothwell,
R.G., 1987.
Sedimentation
on the
madeira abyssal plain over the last 300 000 years.
In
Weaver, P.P.E.,
and Thomson,
J.
(eds), GeotogyandGeochemistryof Abyssal Plains.
Oxford: Geological Society (London), Special Publication,
31,
pp.
71-86.
White, J.D.L., Manville, V., Wilson, C.J.N., Houghton, B.F., Riggs,
N.,
and
Ort, M., 2001.
Settling
and
deposition
of 18t A.D.
taupo
pumice
in
lacustrine and associated environments.
In
White, J.D.L.,
and Riggs,
N.R.
(eds.),
Votcaniclastic
Sedimentation
in
Lacustrine
Settings. Oxford: International Association
of
Sedimentologists,
Special Publication,
30, pp.
141-150.
Cross-references
Bedset
and
Laminaset
Gravity-Driven Mass Flows
Physics
of
Sediment Transport
Planar
and
Parallel Lamination
Sedimentary Structures
as
Way-up Indicators
Sedimentologists
Turbidites
GRAIN FLOW
Perhaps
the
most familiar example
of
a granular material
is the
sand
on a
beach, which
by
itself clearly illustrates
the
dual
solid/fluid nature
of a
granular material. When walking
on the
beach
the
sand easily supports your weight like
a
solid
material.
Yet, a
handful
of
sand will
run out
your fingers
as if
it were
a
liquid.
As it
runs
out, the
same sand will form
a
pile
with sloping sides that shows that material
can
support weight
(at least
up to a
point)
and
thus
has
once again, assumed
a
solid behavior. Granular flow concerns this intermediate fluid-
like movement
of
a material consisting
of
solid particles that
in
other circumstances will behave
as a
solid. This fluid-like
behavior
is
completely
a
result
of
the particulate nature
of the
material. While
the
interstitial space between
the
particles
is
filled with some sort
of
fluid such
as air or
water, there
are
many situations
for
which
the
fluid
has no
significant effect
on
the flow behavior.
The particles interact
by the
elastic deformation
of
frictional
contacts between each particle
and its
neighbors.
The
area
of
these contacts
is a
very small fraction
of
the surface area
of
the
particles
and as a
result
the
bulk granular material will
be
much softer than
the
material that makes
up its
constituent
particles.
Any
force applied
to the
bulk material
is
distributed
between
the
particles
and
supported across
the
contacts.
However,
the
force will seldom
be
distributed evenly. Instead,
networks
of
highly loaded particles known
as
force chains will
support
the
majority
of the
force, while neighboring particles
will experience little
or
none
of the
force, t^orce chains were
first observed
by
Drescher
and de
Josselin
de
Jong (1972).
When will
the
material behave like
a
solid
and
when will
it
behave
as a
fluid? First
of
all,
a
fluid
is
classically defined
as a
material that cannot withstand
a
shear force, hence
a
classical
fluid-like movement
is a
shear motion
(one
with velocity
gradients) that comes about
in
response
to a
shear force.
Imagine
now
that
a
force
is
applied
to the
material, such
as the
weight
of
your foot
or
gravity which makes
the
material
run
out between your fingers. Basically,
if
the particle contacts
can
support
the
force,
the
granular material will behave
as an
elastic solid
and if
not,
the
contacts will break
and the
material
will flow.
But it is not
simply
a
function
of the
magnitude
of
the applied force
as
clearly
a
much larger force
is
applied
by
walking
on a
material than
is
applied
by
gravity
as the
same
material flows from your hand. The difference between
the two
cases lies
in
their different configurations. When resting
on the
beach
the
sand distributes your weight
to the
particles beneath
it. Between your fingers nothing supports
the
weight
and it
flows freely.
To induce flow,
it is
usually necessary
to
first decrease
the
particle concentration
or to
move
the
particle centers apart
increasing
the
interstitial void space. This process
is
known
as
dilatation
and can be
observed while walking
on a wet
beach;
even though
the
sand still behaves
as a
solid,
it
will dilatate
under
the
pressure
of
the footfall which will suck water into
the
increased pore space, causing
an
easily seen local drying
of
the
surface
in the
area around
the
foot.
The
first step
in
inducing
flow is
to
force this dilatation which generally requires doing
work against
the
applied forces. Thus, unlike
a
normal fluid,
one must first force
a
granular material
to
yield before flow
can
occur. This yield determines
the
slope
of
the sides
of
a granular
336 GRAIN SETrUNG
pile which is known as the angle of repose
{q.v.);
steepening of
the slope beyond the angle of repose will cause yield and the
surface material will flow until the slope is equal to or slightly
gentler than the angle of repose.
When static, the majority of the force is carried by force
chains and when the material begins to flow, the particles in
the force chain will move as a unit. In this way, flow is possible
while particles remain locked at the same relative positions to
their neighbors in the chain. This flow state is particularly
likely at large concentrations and for particles with angular
shapes whose protrusions may lock together giving extra
strength to the chain. But even spherical particles can be held
together by frictional forces and flow in the same manner. The
life cycle of a force chain in a shear flow may be likened to that
of a pole-vaulter's pole. The chain is formed as the shear flow
forces particles together, tt will start nearly horizontal, and be
compressed by the shear motion initially generating horizontal
forces. However, the shear motion will cause the chain to
rotate until it is oriented vertically and exerts a nearly vertical
force. Rotating beyond the vertical eventually causes the chain
to collapse. A new chain then forms and the cycle repeats. As
the direction of force rotates with the chain, both shear and
normal forces are generated. Now, the average force generated
depends on the degree to which the chain is compressed which
is largely a function of the particle concentration and in
particular, does not depend on the shear rate. The rate at
which chains are formed is proportional to the shear rate but
the duration of the chain is inversely proportional to the shear
rate so that the product of the two is shear-rate independent.
Thus,
the resultant stresses are shear rate independent or
quasistatic.
Quasistatic flows have historically been modeled using
techniques borrowed from metal plasticity by incorporating
a Mohr-Coulomb failure criterion. Such techniques were
originally used in the Civil Engineering science of soil
mechanics, which describes how granular soils support
buildings. However, in tbat field the concern stops at the
initiation of flow, after which time the foundation fails and the
building falls down. Plasticity models are less successful in
predicting flow bebavior, partially because they generally
assume that the ratio of maximum shear to normal stress is a
constant material property known as the "internal angle of
friction." However, computer simulations have shown that the
ratio is far from a constant.
At lower concentrations (iust how low depends on particle
shape and frictional properties) it is possible to have flows in
which the individual particles are not locked into chains but
can move independently of their neighbors. In an extreme case,
the flow shears so rapidly that, except during collisions, the
particles lose contact with their neighbors and move freely with
a random motion reminiscent of the thermal motion of
molecules in a gas. This is the rapid-flow regime and is
typically modeled using techniques borrowed from the kinetic
theory of gases (see Campbell, 1990). Forces in this regime
must be small as they are carried by the inertia of the particles
and imparted by impact; large forces would require such large
thermal velocities and such strong impacts that particles would
shatter. Furthermore, very large shear rates (of the order of
100 inverse seconds) are required to generate such flows under
Earth gravity. As a result there are essentially no rapid
granular flows of geological importance as geological forces
are large and the corresponding shear rates small. The more
common regime where particles remain in nearly perpetual
contact with their neighbors but still move freely, is not well
understood. However in this case, the stresses are generated
inertially and dimensional analysis dictates that the stresses
must vary as the square of the shear rate. But as the stresses are
inertial, it is again unlikely the shear rates will ever be large
enough to support the large forces encountered in geological
applications. Consequently, most geological flows will fall into
the quasistatic flow regime. The various flow regimes, the
transitions between them, including the many orders of
magnitude changes in the stresses that accompany transition
are discussed in Campbell (2002).
But even when flowing as a liquid, a granular material
retains' some of its solid character. One example already
mentioned is that a granular material must yield before flow
can begin. But also, when confined in a vertical tube, the
friction contacts between the granules and the walls will help
support the weight of the particles. Beyond a certain height, ail
of the remaining particle weight will be supported by wall
friction so tbat tbe force on the material on the bottom of the
column is independent of the height of material above it. (This
result is due to Janssen, 1895, and is one of the earliest
engineering treatments of granular materials.) For this reason,
sand is used to fill an hourglass. Were the hourglass filled with
a fluid, the pressure at the opening and thus the flowrate
depend on the height of liquid above it, making it difficult to
predict the time required to empty the glass. But this
calibration is simple for a granular material, as the pressure
at the opening and thus the flowrate are height independent so
that doubling the amount of material in the glass simply
doubles the emptying time.
Charles S. Campbell
Bibliography
Campbell, C.S., 1990. Rapid granular flows. Annual Review of
Fluid
Mechanics, 22: 57
Campbell, C.S., 2002. Granular shear flows at the elastic limit. Journal
of Fluid Mechanics, 465: 261-29t.
Drescher, A., and De Josselin de Jong, G., 1972. Photoelastic
verification of a mechanical model for the flow of a granular
material. Journalof the Mechanics of
Physics
and Solids, 20: 337.
Janssen, H.A., 1895. Versuche uber getreidedruek in silozellen. Z.
I4!r.
Dt. Ing., 39: 1045.
Cross-references
Angle of Repose
Debris Flow
Grain Size and Shape
Gravity-Driven Mass Flows
GRAIN SETTLING
Pebbles settle faster in water than do grains of sand, and any
particle settles more slowly in a viscous fluid such as oil. An
understanding of grain settling is important in analyzes
sediment transport and deposition. The turbulent eddies of
flowing water in a river or ocean current lift sediment particles
above the bottom and transport them in suspension. This
CRAI.N SETTLING
337
GRflIN DIAMETER. D in
Figure G19 Curves for the settling velocities ot sphericil grains having
densities that range from quartz (2.65g/cm'i to gold il8g/cm'). The
dashed curve is for the settling of naturdl sand grains js measured by
Babd and Komar
(19811,
giving some indication of the iniporlance
of the grain's nonsphericity on its settling velocity.
process depends on the settling velocities of the grains, the
lower their settling rate the higher the grains are lifted above
the bottom and the greater their concentration. As the strength
of the current wanes, the grains are deposited on the bed at
rates that depend on their settling velocities.
A nutnber of studies have been conducted to tneasure
settling velocities of sediment grains, with the anaK/^es relating
the settling to the sizes and densities of the grains. Such
measurements are usually tnade in a cylindrical tube filled
with water. When the grain is released in the water, it initially
accelerates., increasing in velocity until a balance is reached
between the immersed weight of the grain in water, which is
(he net downward force causing it to settle, versus the drag
force produeed by the friction of the water that resists the
grain's movement. When a balance is achieved between
these opposing forces, the grain settles at a constant velocity,
its terminalvelocity. It is this terminal velocity that is tneasured
in settling-tube e.xpcriments, and used in most applications.
Equations can be derived for the prediction of this terminal
velocity by balancing relationships for the immersed weight of
the grain and lUtid drag. If
one
considers a spherical grain with
a diameter D and density />,, its itnmersed weight is equal to
ihe difference between its density and that of the
fluid,
p, times
the acceleration of gravity g. and the volume of the sphere,
giving (p,-p)g(nD^tb). For small particles settling at a slow
rate.
Sir George Stokes derived a relationship for the fluid
drag,
equal to 37r/(/)ii\ where /(is the fluid's viscosity and ir, is
the settling velocity. If the force of drag is equated to the
itnmersed weight and then solved for ii\. one obtains
18/f
the Stokes settling equation. It is seen that in the Stokes range
the settling velocity of a sphere depends on the square of its
diameter, the density difference between the grain and
fluid.
settling of spherical
- grains(CSF
=
woter of
SETTLING VELOCITIES IN
WATER OE QUARTZ DENSITY
NON-SPHERICAL GRAINS AS
A FUNCTION OF THE COREY
SHAPE FACTOR (CSF)
sand sand
' I I I
1
I I
10"'
I
NOMINAL GRAIN DIAMETER,
Figure G20 C^urves for ihe seltling of particles having the shape of a
triiixicil ellipsoid, with the nominal diiimetcr being that of an
equivalent sphere having the same volume and v^'eijjht. The settling
velocity depends on ihe Corey Shape Factor
(CSF),
a measure of the
grain's sphericity. After Komar and Reimers (1978).
and is inversely proportional to the viscosity of the
fluid.
Application of
this
equation is limited to values of
the Reynolds
number.
Re
= jiDwJft. that are less than about 0.5, that is to
small cotnbinations of D and u\.. For quartz-density spheres
settling in water, this limits its use to particles of diameter
less than about 0.08
mm.
that is primarily to silt and clay. For
large particles that settle at high velocities (pebbles), the drag
becomes proportional to the square of the velocity. The
derived relationship for the terminal velocity is then propor-
tional to \/D. The relationship for the settling of large particles
is also complicated by the inclusion of
a
drag coeflicicnt that
must be evaluated from empirical curves that are based on
actual measurements of settling spheres.
Curves of settling velocities in water versus grain diameters
are given in Figure G24 for a range of sediment grain densities,
from quartz
(;J,
= 2.65g/cm-') through the range of comtiion
heavy minerals, and including native gold with its very high
density of I8g/cm\ As
expected,
for a given grain diameter the
denser the tnineral the higher its settling velocity. The cut-off
Reynolds number Re-0.5 tor application of the Stokes
equation is shown in the diagratn. In the Stokes ranee of
small particles, the increase in ii\ is proportional to D'\ the
proportion to \/1) does not occur for quartz-density grains
until the diameter is on the order of lOnim. There is a zone of
transition,
which itnlbrtunately in the case of quartz-density
338
GRAIN SIZE AND SHAPE
grains corresponds to diameters in the sand-size range, making
it more difficult to evaluate their settling velocities.
The curves in Figure G19 are for the settling of perfect
spheres. Most sediment grains arc not smooth spheres, so the
effects of particle shape (sphericity and angularity) must be
considered. The dashed curve in Figure G19 for natural sand
grains is based on the measurements by Baba and Komar
(1981) relating the terminal settling velocities to the inter-
mediate axial diameters of the natural grains whose overall
shapes are roughly that of a triaxial ellipsoid so the intermediate
diameter is representative of the grain's overall size. The
resulting curve is seen to be at lower settling velocities than
the curve for quartz spheres, since the nonsphericity and
angularity of the natural grains increases their drag and
thereby reduces their settling velocities.
Experitnetus have been eonducted on the settling of
regularly shaped but nonspherical grains, such as perfect
triaxial ellipsoids or cylinders. Curves ba.sed on experiments by
Komar and Reimers (1978) with triaxial ellipsoids are given in
Figure G25. with each curve depending on a value of
the
Corey
Shape Factor
CSF =
D,
Bibliography
Baba,
.1..
and Komar, P.D., 1981. MeasuremenU and analysis of
settling velocities of tialttral quartz sand grains. Journal of
Sedimentary
Petrology.
51: 631-640.
Berthois, L., 1962. Etude du eomportement hydraulique du mica.
Sedimcniology.
1; 40-49.
Dietrich.
W.E.. 1982. Settling velocily of natural panicles. Water
Resources
Re.search.
18: 1615-1626.
Kajihara. M.. 1971. Settling velocity and porosity of large suspended
particles, Journalof
the
()(cano^rapliical
Societv
ot
Japan.
27: 15S-
162.
Komar, P.D.. and Reimers. C.F... 1978. Grain shape cffL-cts on settling
rates.
Journal
of
Geolo^v.
86: I9.V209,
Komar. P.D.. Morse. A.P.. atid Small. L.F.. 1981. An analysis of
sinking rales of tiatural copepod and euphausiid fecal pclleis.
Limnology and
Oceanography.
26: 172- 180.
Komar. P.D.. Baba. J,. atid
Ctti.
B.. 1984. Grain-si/e analyses of mica
within sediments and the hydraulic equivalencu of mica and quartz.
J<nn-nal
of Sedimentary Petrology. 54: 1379-1391.
Cross-references
Flocculatlon
Grain Size and Shape
Hindered Settling
where
D,,. D,..
and D, are respectively the longest, intermediate
and shortest axial diameters of the triaxial ellipsoid. This shape
factor evaluates the degree of departure frotn a sphere, which
has a value CSF= 1. the greater the departure the stnalter the
value of
CSF.
The series of curves in Figure G20 is seen to
depend on CSF. progressively shifting to reduced settling
velocities cotnpared with the curve for perfect spheres as the
value of C^'f decreases and the grains become less spherical.
Although the natural sand grains employed by Baba and
Komar (1981) in their experiments, yielding the dashed curve
in Figure G19. where not perfect triaxial ellipsoids, the degree
of departure of the measured settling velocities from the curve
for perfect spheres was still found to depend in large part
on CSF.
The extreme case of particle shape affecting settling occurs
for flat mica plates. Experiments have been undertaken with
some success by Komar
etal.
(1984) to measure settling
velocities of mica and to relate them to the grain's degree of
flatness (the ratio of the thickness of the plate to its average
two-dimensional diatneter). However, the analysis of tnica
settling can be complex due to the oscillations they sometimes
undergo, particularly for larger trtica plates (Berthois. 1962).
The degree of angularity or roundness of the grain can also
have some effect on its settling velocity. However, grain
roundness has been found to be less important than sphericity
(Dietrich,
1982).
While most applications involve assessments of the settling
velocities of silt and sand grains, there is also interest in the
settling of clay, which is complicated by its tendency to form
low-density floes, the density of which often decreases as the
sizes of the floes increase (Kajihara. 1971). Another applica-
tion is to the settling of fecal pellets and shells of organisms
such as foraminifera, important to deep-sea seditnentation.
Fecal pellets are commonly elliptical or cylindrical in shape,
and measurements of their settling velocities have been shown
to agree with equations for the settling of such particle shapes
(Komar
(^/(//..
1981).
Paul D. Komar
GRAIN SIZE AND SHAPE
Introduction
Size and shape are conceptually distinct properties of
sedimentary grains. The fortner deri\es from reference to an
external metric to determine the absolute magnitude of
a
grain,
while the latter represents relative measures of the grain to
establish its geometry. In practice, the two are related, first by
the means that arc customarily employed to establish the size
of
a
grain, and second by the fact that grain shape in a sample
might vary systematically with grain size. Size and shape of
constituent grains are component properties of the texture of
granular aggregates. Most grains originate by disintegration
of bedrock, so size and shape are set initially by characteristics
of the rock and by the disintegration process. Grains may also
be formed by chemical precipitation or by cementation of
previously existing grains, whence the initial properties are set
by the endpoint of those processes. Size and shape are
subsequently tnodificd in the surface environrnent by weath-
ering and by the damage imposed by processes of scditiient
transport.
Size and shape are fundamental properties of clastic
sediments. The distributions of size and shape in sediment
deposits influence and index other important physical proper-
ties of
the
sediment, such as porosity, permeability, and surface
roughness, they carry important information about the origin
of a deposit, they affect the stability of the deposit, and they
influence habitat qitality for stnall organisms. In this article we
first examine conceptual and operational tneasurcs of grain
size and shape. We then consider sampling procedures for
determining representative size and shape in sedimentary
deposits, since sampling places some constraints on the
information that practically can be had. Finally, the distribu-
tions of size and shape properties of sedimentary aggregates
are considered. A more extended review is given by Pye (1994).
GRAIN SIZF ANDSHAI'f
339
while the classic work on satnpling and analysis of sediment
properties remains that of Griffiths (1967).
Grain size
The size of a compact, three-dimensional object such as a
sedimentary grain tnight be indexed by some measure of its
volume, or by some linear rneasure of its geometry. For
geometrically regular objects, either carries equivalent infor-
mation.
For irregular objects, including sediment grains, they
do not. Both approaches have been used for characterizing
grain size. Conceptually, the most thorough measure is
provided by grain volutne. or by the equivalent linear tneasure
\/l'. The most comrnon explicit rneasure, however, is the
notional length of the second principal axis
(^-axis)
of the
grain (see Figutc G21). This tneasure gains its prominence
from the operational method for sizing grains by washing or
dry passaee through a sequence of
sieves
with defined openings
(DIN mi}. 1983: BS8l2-ia3. 1985: ASTM D422-63, 1990:
D75-87.
1992) It is always possible for the c/-axis to pass
through on end, so grain size is determined by the /i-axis
length.
Sieves with round opetiings gauge the A-axis accurately, but
square-tnesh screen sieves are custotiiarily used. Passage
through a square opening depends jointly on h- and c-axis
ditnensions of the gtain (Figure G2I). so the measure is biased
in comparison with a caliper measurement. The sieve opening,
D. is customarily accepted as the practical measure of grain
size.
This technique is commonly applied down to 0.063 mrn
(63 pm). the lower litnit of sand sizes, but the dry technique is
inefficient below I23|.tni. Si/e detertnination in this tnanner is
in fact a classification procedure: sizes are determined only to
within the litnits established by successive sieves. The conven-
tional classification scale is a geometric progression with ratio
2.
known as the Udden-Wentworth scale (Table G13. which
includes the equivalent (/)-scale employed by sedimentologists).
hi research practice, clasts larger than about 64 mm cotiimonly
are measured individually using a ruler or calipers (usually
without bias correction for D). although grains up to boulder
size tnay be classified through screens in industrial operations.
Very small grains defy efficient size separation by screens.
Two methods dominate size determination of fine materials.
The traditional tnethod entails particle settlement in water.
The density of an aqueous suspension of particles is
periodically measured, traditionally, by pipette withdrawal or
hydrometer (BS34()6-2. 1984: ASTM b422-63. 1990). but in
tnodern methods by optical or X-ray beam transtiiission. The
limit of sizes that have cleared the measured suspension or the
beam path at the time of measurement is determined from
Stokes' Law\
(Eq.
in which r, is the terminal fall velocity of a spherical grain in
the latiiinar regime, g is the acceleration of gravity. Pv and p
are respectively the solid atid fluid densities, and i} is the
dynarnic viscosity of the
fluid.
The notiotial grain size assigned
by this technique is the diatneter of a sphere settling in still
water in the laminar regime at the same rate as the observed
particle (the nontinal or
settling
diameter).
Stokes Law holds
(i.e..
the settling regime is satisfactorily latninar) for grains
liner than 5()).ttn in sitspcnsions more dilute tiian about
0.95
0.90
0,85
0.80
0.75
0.7(;
0.2
0,4 0.6
c/b
0.8 l.O
Figure
C21
The ralin of
square
sieve opening
[D]
to
b-axis
length
as <i
tuncti(.)n ot (last flatness,
c/b.
D/h
gives the bias between commonly
enipkiyed measures ot grain
size,
and is the correction factor required
to convert the caliper-dotermined 6-axis to the equivalent square sieve
opening,
c must be measured lo calculate it. Insets: 1. definition of
principal axes of
a
grain; 2. Geometry of
a
gr.iin in 3 square mesh
opening.
1 percent. The practical lower litnit of size determination by
settling is around 0.5).tm. below which Brownian effects
dominate the settling behavior. A similar technique can be
used to determine the settling diameter of sands (USFIASP.,
1957) but. in this case, tertninal velocities fall in a transitional
region between laminar and fully turbulent settling
In settling techniques, the actttal grain size of the irregular
particles is nearly always different than the settling diameter.
The assigned grain size is no longer sitnply related to particle
geotnetry at all (Figure G22). The tneasure is nonetheless
convenient since, in seditnentology. the settling behavior of the
grain in water is often a critical element of interpretation.
Hallermeier (1981) presented empirical functions lor the
settlement of natural quartz grains and Dietrich (1982)
presented generalized functions for settling velocity that
subsutiie vat"iations in size, density, shape and roundness.,
while Komar and Cui (1984) have systematically studied the
variation between geometrical diameter (/i-axis) and the
measures realized by sieving and settling methods (see Grain
Settling).
A modern technique for determining the size of fine grains
entails the measurement of laser or optical beam interruption
intervals induced by a suspension pumped through the beam
(BS3406-7. 1998). This technique measures an arbitrary
diameter or "cut" across the grain. The relation between the
distribution of random cuts and the true diameter for grains of
fixed shape is known for dilute particulate phases (Nicholson,
1970). and is used to derive a size distribution.
The methods for sizing grains are ahnost exclusively applied
to aggregations of grains to return a frequency distribution of
sizes in the aggregation. The size of individual particles is not
distinguished. If individual sizes are of interest, they can be
340
Table
C13
millimeters
The
Urlden-Wentw(>rth scale
Size
of
squared opening
tor classifying
micrometers
GRAIN
grain size
SIZF
,
with
AND SHAPE
.^-equivalents'
U.S. Standard/ASTM
''
sieve mesh no.*"'
Size
description
class
d
4.096
2.048
1.024
512
256
128
64
32
16
8
4
2
I
0.5
0.25
0.125
0.0625
0.0312'
2.000
1.000
500
250
125
62.5
31.2"
15.6'^
l.r
3.9'-'
-12
-I!
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
10 in.
5 in.
2 1/2
in.
I 1/4
in.
5/8
in.
5/16 in.
5 (mesh)
10
18
35
60
120
230
Very large boulders
Large bottlders
Medium boulders
Small hoiilders
Large cobbles
Small cohhles
Very coarse gravel
coarse gravel
Medium gravel
Fine grave!
granules (very ftne
gravel)
Very coarse sand
coarse sand
Medium sand
Fine sand
Very tine
s;ind
coarse silt
Medium silt
Fine silt
Very fme silt
" A comprehensive churt nl grain sii^e scnks is gucn b\ Truesdell and Vanics
11
ySOi,
^ <p- -log;/?, whcrt- P is the assigned grain Jiamctcr in mm, (The minus sign wus introduced so thai sand sizes would have pusilivc (^-numbt-rs. Gravel
sedimentologists today oflen ignore the minus sign and use
iji
log^tJ = -</'-)
'' Approximate mesh opening size (inches) or number of mesh openings per inch.
•^ Terminology established by Wcntworth (I'J22|. Modified in modern useage,
' Nominal diameter: usually analyzed by settling rale in Huid.
' In certain classilicaiions, the upper limit of clay sizes is 2 microns, Texlural usage is noi related lo the mineralogical useage of the term "clay".
determined by the use of calipers, by tracking individual
particles in a settling column (practical down to about 125|.tm
without special arrangetnents), or by exatnination of grains on
a microscope stage. Particulate floes complex aggregations of
fine particles that may include organic material present
special problems. Floes commonly are open, porous structures.
The most practical method that has been forwarded for sizing
them is to make mierophotographs of individual grains in a
settling eolumn or to deposit them on a lilter or mieroseope
slide and to analyze the photographs using image analysis
teehniques (BS3406-4. 1998). The eomplex shapes of sueh
grains make ordinary linear measures of size relatively
meaningless. Shape eotnplexity of sueh grains has been
measured by fractal analysis (Logan and Kilps. 1995).
There has also been substantial interest over the years to
obtain grain size measurements frorn photographs. Either
grains are actually measured on the photographs, or they are
counted and correlated with mean or median diameter using a
calibration curve constructed from several photos for whieh
there are corresponding field tneasuretiients. The results vary
with grain shape, paeking and characteristic attitude. For the
case when the grains are essentially flat lying (i.e.. lying in
the principal axis, or ab plane), unbiased distributions and
summary statisties tnay be reeovered for a truncated range
with lower cutoff eorresponding with the practieal limit to
resolve grains in the photo. When grains are not consistently
oriented, the photo view amounts to a sequence of random
euts through the grains. The situation is similar to that
encountered in beam interruption methods, but the grains are
1.0 r
•— Laminar Region—*
: pj =2,700 kg m-3
: PI
=
997
kg
m
3
- T = 29°C
. X. =9.11
X
lO^Nsm-^
"-Transition Region-*
•$v
/
i' ^/
r
6-Turbulent Region-*
^^
Observations ±25,^1
0.1
1
0
Grain size (D, mm)
10
Figure
G22
Settling diameter
of
grains compared with
D,
determined
by sieve classification. Note thai
the
observed results (crosses detined
by error bars, derived from many experiments
in a
large settling
cylinder) depart systematically from those predicted
tor
spheres with
diameter
D. The
difference
is
mediated
by the
irregular shapes
of
naiura! grains, vi/hich means bath that
D is a
biased measure
of
equivalent spherical diameter
and
that
the
grains adopt complex
settling trajectories
in
water.
GRAIN SIZE AND SHAPE
tiot tiow dispersed. Kellerhals etal. (1975) made
a
detailed
study
of
this situation (which
is
analogous
to
grains
on a
thin
scclion)
by
niiiking initncrical trials
on
ellipsoid shapes
and
found thai
the
measured a-axis elosely estimates
the
true
/'-a.\is.
atid that
the
tneastired /'-axis eiosely estimates
the
true
i-axis. Krunibein (1941) toLind that
ati
ellipsoid generally
provides
an
aeeeptable approxitnatJon
of
aetual grain shape,
so
the
results
of
Kellerhals etat. appear
to be
robust. However.
most deposits
of
elastic grains possess some preferred fabric,
whieh would render
the
result invalid.
The
methods discussed
in this paragraph
are
under renewed scrulinv
now as
modern
image analysis techniques promise automation
of
what
has
been
a
tedious probletn (Butler etat.. 2001).
Grain shape
Measures
of
grain shape
may be
divided into
two
distinct
categories: ones which seek
to
express surface curvature
or
stnoothness.
and
ones which seek
to
represent
the
overall shape
or biilkiness
of the
grain.
Measitres
of
surface "smoothness"
or
routidness
are, in
effect, treasures
of
grain wear
in the
surface environnieni.
A
iiLtmber
of
tneasures have been proposed,
all
related
lo ihe
ratio betweeii
ihe
niinitniim radius
of
curvature over some part
of
the
grain surface
and
sotne measure
of the
overall grain
diameter (Figure G23): that
is.
R
=
2r,,,,Ja
or R =
Ar,,,,,,l(a
+ h)
(Eq.
2)
where
r,,,,,,
is the
tninitnittn radlits
of
curvature.
For
grains
larger than
a few
tnillitiieier
in
diatneter,
the
tneasurenients
are
eonventionally tnadc
in Ihe
prtneipal
a.\is
plane.
The two
quoted measures differ otily insofar
as the
seeond averages
the
reference diameter
of the
grain over
two
principal axes.
The
range
is
0</?<l,
the
limits being represented
by a
grain with
a
sharply broken corner (r,,,,,,
-• 0) and a
circle
or
sphere.
Ratios
of ihe
prineipal
a.\es
represent elementary tneasures
of grain shape. Grain elongation
(F)
tnay
be
represented
by the
ratio
atb. and
grain "flatness"
by the
measure
ctb.
Other
measures
of
tlatness inelude F=2cHa + b),
and C=
ct{ati)^'~,
the Corey Shape
Factor.
Elongation
is an
open-ended measure
with minimutn value
1. It has
proven useful
for
identifying
grains suitable
for in
situ orientation measurements (various
investigators specify E>312
or E>2 for
such measurements).
Zingg ([9?'5) inverted Eand used
the
axial ratios hta
and
ctb
lo
deftne
a
two-parameter description
of
grain shape.
He
deliiied
four characteristic shapes
on the
basis
of
these ratios
(Figure
G24)
which have proven useful because they reflect
well
the
telative mobility
of
grains
in
iraetion utider fluid shear.
Krumbein (194!) observed that grain shape
is
fundamen-
tally expressed
as the
ratio
of
grain surface area
to
surface
volutne. Nondimensional measures
of
grain shape
or
bulk
mighl logically
be
based
on
reference
to
sotne ideal
or
limit
shape,
e.g.,
{/V/V^,..
in
which
V.,,
is the
volume
of
the smallest
circLunscribcd sphere
(the
sphere with diatneter
a). For
small
graitis, this quaiitily
is
difticuit
to
tneasure.
so
alternatives
include ^. .4M,,.
in
which
A is
grain surface area,
or
projected
area
in the
principal axis platie.
and A.., is the
area
of the
smallest circumscribed circle,
or
simply
the
axial ratio
bta or
2bl{a+h).
Wadell(1933)adopted
an
alternative definition/I
vM.
in whieh
A, is the
surface area of a sphere with
the
same volume
as
the
given particle.
The
ratio
is
itiverted with respect
to the
previous suggestions
in
order
to
preserve
the
limited range
[0,1].
Ihis proposal
is
rational,
but the
measuretnent
is
tedious.
Figure G23 llluslrtititig the prltuiple ol' a measure of smoothness or
roundness ot n grain. By (onvention, radii (.reatt'ci by individual
mineral asperities on the grain surface are ignored (since they may
reflect varying mineral hardness rather than overall wear).
Disc
Sphere
(Equant)
0
0.1 02 O.I 0.4 0.5 0.6 |0.7 08 0.9 1.0
c/b I
2/3
Zingg boundaries (particle shape)
Blade Rod
Figure G24 Zingg grain shape diagram, including Krumbein inlertept
spheritities.
Accordingly. Krutnbein (1941) introduced intercept sphericity
as
a
close appro.xitiiation
of
Wadell's measure:
S =
^he/a~.
The quatitity under
the
surd
is the
ratio
of the
volume
of the
approximating ellipsoid
to the
volume
of the
eircutn-
scribing sphere. Sneed
and
Folk (1958) proposed nutxitnutn
projection
.spherieity.,
4^
—
{/(c-/ah)
—
C'-''', noting that this
measure represents
the
ratio
of
maximum projeeted areas
of
a
sphere
of
equivalent volume
and of the
partiele, hence
is a
rational measure
of the
effect
of
particle shape
on
settling
velocity. This jitstilication accounts
for the
wide
use of T and
C
in
studies
of
clastic scdimentatioti.
342
GRAIN SIZE AND SHAPE
Sampling grains
Sampling enters into the definition of grain size because, in
most instances, grains are collected and sized in aggregate
(such as by sieving), so that a distribution of sizes is obtaitied.
Grains are not measured individually. To the extent that the
sampling procedure biases the selection of grains, it influences
the observed results. For example, point (grid), linear, areal
and volumetric methods for selecting tnaterial predetertnine 0.
I. 2. and all 3 dimensions, respectively, of the sample volutne.
whilst number and weight as bases for assigning frcqttency to
separated size classes also vary by a factor O. Kellerhals and
Bray (1971) presented a formal conversion procedure based
upon the dimensions of
a
sample that are predetermined by the
sampling and analysis techniques so that observed results can
be represented in terms of any preferred measurement
conventions. The general conversion formula is
(Eq. 3)
in which /,„ is the observed frequency, D, is mean grain size for
size class /, .v is the integer dimension required for the
conversion. // is the number of classes, and Ihe sum effects a
renortnalization of the distribution to yield the converted
frequencies/,.,. Table G14 gives the dimensions for conversions
amongst any usual combination of sampling atid frequency
assignment methods. For the conversion grid-by-numbcr (the
usual means to select and analyze individual grains frotn
outcrop or surface) to volume-by-weight (classifted bulk
samples), .v - I: these two standard procedures are equivalent
in principle. Photographie samples are area-by-number satn-
ples.
for which the conversion factor to equate ihe result wilh
standard analyzes is
A-
=
2.
A number of subsequent analysts
have criticized the Kellerhals Bray procedure on the grounds
that the sitnple paeked cubes model employed by Kellerhals
and Bray to illustrate their concept is defective since it does not
realistically model the porosity or grain arrangements in real
earth tnaterials. It should be etnphasized that the formal model
does not depend upon the illustration at all. Properly
conducted tests (Church etal.. 1987) confirm the efficacy of
the formal conversions.
Counts are used to establish frequency by size of grains
when relatively small numbers or comparatively large grains
are selected from outcrop or a sedimentary surface, and when
grains are selected on a microscope slide. They would normally
be seleeted on a grid or by application of randomly drawn
coordinates (see. for example. Wolinan, 1954). Weights are
Table G14 Weighling factors for conversion of grain size frequency
distribution:;''
Conversion to
sieve by Grid by Cirid by Area by Area by
Conversion from weight tiumber weight tiumber weight
Sieve bv weight 1
Grid by tuitiiber 1
Grid bv weight D "
Area by tiumber D^
Area by weight D "'
1
1
D •
D"
D '
D'
D'
I
D"
D-
D ^
D -
1
D '
D
D
D -
D'
1
.^ftcr Ktfllerhyls and Brj>
(1971.
their table 2),
reeovered when large numbers of grains are sorted in sieves,
and when settling methods are used. An important distinction
in the quality of the information in these two cases is that
number counts constitute legititnate quantities for statistical
manipulation: frequencies derived from weights do not.
Knowledge of grain size, paeking and tnineral density tnake
the derivation of approximate grain nutnber equivalents
feasible but. in practice, the numbers become so large for tine
materials that no useful statistical resolution may be gained
anyway.
The question arises how large must a sample be in order lo
obtain an unbiased representation of sizes, or an estimate of
some summary measure—sueh as the mean size- to within
some desired level of precision. For samples in whieh fre-
quency is assigned by enumeration (typically, grid-by-nutnber
samples—cotnmonly known as "Wolnian samples"), these arc
statistical questions that have been considered by many
analysts. Table GI5 summarizes some results.
Frequency is assigned in a bulk sample by relative weight, so
the numbers of grains present in one size class influences the
observed frequency in all other classes. The critical class is that
whieh contains the largest grains, since these will be the least
numerous in comparison with their aggregate weight. One
requires a sample large enough to ensure that this class is
adequately represented. The requisite weight for a given
maxitnum grain size is the subject of several standards (ISO
4364.
1977; ISO 9195, 1992; ASTM D75-87, 1987; BS3680-
lOC.
1996; BS3680-1-E, 1998). In the sedimentary literature.
Church etat. (1987) atid Milan etal. (1999) have pursued the
question of adequate sample size for gravels. Church et al.
determtned that the largest gr;tin should not constitute tnore
than about O.I percent of the sample weight in order to obtain
a reliable analysis. The criterion is not conservative (compare
ISO criteria, based on the D^^ size: Figure G25). but it
nevertheless leads to sample sizes that arc impractical for
manual sampling of material coarser than gravel. Ferguson
and Paola (1997) have estimated by numerical simulations the
bias and precision of percentile estimates in bulk si/e
distributions. Their results show that, if only information
about relatively central measures is required, then substantially
smaller samples are tolerable than those suggested for
characterizing the full distribution, and that results depend
upon the gradation (range of sizes) of the material.
There remains the additional sampling question raised by
the probability that the sampled materials are inhomogeneous
over the satnpling surface or volume. If one is interested only
in average measures of the material, it is far more convenient
to sample over the variability than to attetnpt to identify and
satnple each homogeneous subunil (which may be very difficult
to identify in some circumstances). Wolcott and Church (1991)
and Crowder and Diplas (1997) have addressed this question
for bulk and surface sampling, respectively. A speeial probletn
presented by fluvial gravels and by sotne olher deposits is that
the surface is characteristically different than the subsurface
because of the loss of fmes from the surface by settling or by
winnowing (armoring). Consequently, surface (by point, linear
or areal means) and subsurface sampling (by volumetric
means) are customarily conducted separately in coarse
materials with evident segregation.
Because the distribution of grain sizes in a deposit may be
very wide (possibly surpassing five orders of magnitude), so
that it may not be feasible to use a single technique to sample
or analyze the entire range of sizes that oeeur. and because
Table G15 Criteri.i for
Authority
Wolman (1954)
Hey and Thome
(1983)
Moslev and Tindaic
(1985)
Church <•?<//. (1987)
Fripp and Diplas
(1993)
Rice and Church
(1996)
GRAIN SIZE AND
j;ri(l4iy-number sample size'
Basis of recommendation
9 replicate samples
Operator variance
Cumulative histogram
Statistical theory
Binomial theory
Bootstrap analysis
SHAPE
Parameter
median
mean
mean
Precision of the
mean
Percen tiles
Percentiles and size
distribution
Recommendation
100 iirains
100
70
Variable
200 400
400
" MainK after Rice
and
Church
4
810 100
b
-
axis
of
largest stone
(mm)
1,000
Figure G25 Bulk sample size standards based nn the larj^est grain
constilutingO.I percent of the total sample weight (after Church ef a/.,
19H7;
Milan et JI., 1999). ISO 4364-I977lE}, hased on Ihe Da.,, is
for comparison.
satnpling requirctncnts
may
become unrealizable
for
deposits
thai include very Uirgc materials, samples often represent only
a portion
of
the entire size distribution.
The
satnple
may
still
contain signilicani infonnation
(as,
e.g.. wheti
it is of
interest
lo satnple the matri.\ materials
in a
diamict sediment). Because
frequeticies
are
assigned
on a
relative basis,
it is
important
to
constrain comparisons between satnples
to the
comtnon range
of aiialyzed sizes. Such samples
may be
regarded
as
truncated
or censored samples
of
the entire si7:e distribution. Combining
data from different sampling techniques tnay require adjust-
ment
of the
assigned sizes
in one or the
other portion
of the
range,
as the
consequence
of the
nonequivalence
of
size
assignments amongst different techniques (Komar
and Cui,
1984). The most frequent example
of
this problem arises
in the
joining
of
sieve data
of
sands with
the
data
of
silts derived
by
settling tnethods.
The distribution
of
grain sizes
The distribution
of
grain sizes arrived
at by
enumeration
is a
statistically valid distribution,
but the
distribution
of
sizes
arrived
at by
weights
is
not. Sinee
the
relation between
the two
bases goes
as D .
frequency
by
weight represents
a
nonlinear
transform
of
frequency
by
number. Despite
the
ill-conditioned
nature
of
distribution
by
weight, sedimentologists have long
used mathematical functions
lo
approximate grain size
distributions (GSD).
Tbe
procedure
is
essentially descriptive,
and
no
rigorous goodness
of fit
test.^;
are
available
for
comparing distributions with data
or
with each other. Analysis
is characteristically conducted
in 0
-space, since
the
logarith-
mic transform
is
necessary
to
hotnogenize the variance
of
grain
size data.
Two distributions dominated elassical thinking about CiSD:
the so-called
Rosin distribution
(see
Kittletnan, 1964) which
is.
in fact,
the
two-parameter Weibull distribution,
and the
tog-
normal distribution (Krumbein. I9^S), both customarily
applied
lo GSD by
weight.
The
former
is
Justified
on the
basis
of the
size distribution
of
crushed tnaterial (Bennett,
1936;
see
also Turcotte, 1992), whilst Inman (1949) atid
Middleton (1970) have found justifications
for
log-normal
distributions
on the
basis
of
sediment sorting
in
transport.
The
Rosin-Weibull distribution
has
been suecessfully applied
to a
range
of
naturally broken tnaterials.
but
tnostly with
•O,,,,,v<32mm (^>-5). Exceptions exist,
but a
plausible reason
for an upper size limit
is
thai
the
size
of
larger "broken" grains
will almost always reflect primary jointing and bedding planes
at outcrop, which are often
not
random. Ibbeken (198.'^) gave
a
review
of
empirical results.
The
log-normal distribution
appears
to
describe tolerably well
a
wide range
of
sedimentary
materials,
but it
often
has
been adopted without critical
analysis
as a
matter
of
descriptive expediency.
In recent years, additional distributions have been proposed
to describe transported sediments, including
the
log-hyperbolic
344
GRAIN SIZE AND SHAPE
distribution and the log-skcw-Laplace distribution. Both arc
related to the nortnal distribution (the hyperbolic distribution
being the most general). Fieller etat. (1992) give a brief but
comprehensive description of the application of distribution
functions to describe GSD.
Many GSD arc clearly not simple. Bitnodal or nutllitiiodal
distributions have been described using mixture distributions.
However, insofar as multiple modes usually signify sub-
populations that have been intlueneed by distinctive processes,
il seems most sensible to partition the distribution into
unimodal components and to study each one. The partition
procedtire. formerly tedious, is easy today using algorithms
designed for this purpose (whieh usually require that the
characteristic distribution be specilied).
In the past, GSD have customarily been surnmarized by
statistics, usually the first three moment statistics or graphical
approximations of them. Approximate moments based on
t^-percentiles have been advocated (Inrnan, 1952; Folk and
Ward,
1957) and have the advantages that their consistent
interpretation does not presuppose a particular distribution
function,
nor a complete expression of the distribution
function (the extreme tails often remaining poorly sampled
or unknown in sediment analysis), or a statistically valid
description of the distribution. Strenuous efforts have been
expended to associate mornent domains with particular
depositional environments, which implicitly supposes that
some characteristic GSD is associated with an environment.
Whilst some general associations were recognized the project
did not identify clean discriminations and is not now much
pursued.
Distributions
of
grain shape
The distribution of shape characteristics has received much less
attention. A significant issue is the definition of a homo-
geneous population of grains for consideration, sinee shape
variations may systematieally be alTected by grain size and by
grain lithology. Shape measures have mainly been studied in
terms of changes in median or tnodal values along sedimenta-
tion gradients. No general distributions have been proposed
and there appears little reason to expeet any. in homogeneous
samples, beyond normal or log-normally distributed random
variations.
Miehael J. Church
Bibliography
ASTM (American Society
for
Testing
and
Materials),
1992.
D75-ii7:
standard practices
tor
sampling atigregates.
ASTM lAmcrican Society
for
Testing
and
Materials).
1990.
D422-63:
Standard tcsl method
for
particle size analysis
of
soils.
Bennett, J.G., 1936. Broken
coal.
Journal
ofrhe
Institureof
Fuel.
10:
22
39.
BSI (British Standards Institution).
1985.
BS8I2-I0.3. Methods
for
dctcrinitKitioti
of
particle size distribiuion.
BSI (British Standards Instiltition). 2(100. BS34()f)-2(
1984).
Recom-
mendations
for
gravitational liquid sedimentatioti tnelhods
Tor
powders
and
suspensions.
BSI (British Standards Institutionl.
1998.
BS34U6-4(I993). Guide
to
tnicroscope
and
image analysis methods
(1
^im
to
1
mm).
BSI (British Standards Institution).
1998.
BS.M()6-7n995). Recom-
mendations
lor
sintile particle liulit interaction methods.
BSI (British Standards Institution).
1996.
BS368()-1I)C. Guide
to
methods
ol
sampling sand
bed and
cohesive
bed
materials.
BSI (British Statidards Institution}.
!998.
BS368()-IOE. Sampling
and
analysis
of
gravel
bed
material.
Butler,
J.B..
Latie,
S.N.. and
Chandler.
J.H., 2001.
Autotnated
extraction
of
graiti-size data from gravel surfaces using digital
image processing.
Journal
of Hydraulic
Research.
39: 519 529.
C'luirch"
M.A..
McLean.
D.G.. and
Wolcott.
J.F.. 1987.
River
bed
gravels: sampling
and
analysis.
In
Thorne.
C.R..
Bathursl.
J.C..
and
Hey, R.D.
(eds.). Sediment
Transport
in
Gravel-hed
Rivers.
Chichester: John Wiley
&
Son.s.
pp. 43 88.
Crowdcr,
D.W.. and
Diplas.
P.. 1997.
Sampltng heterogeneous
deposits
iti
gravel-bed streams. Journal
oJ
Hydraulic
Engineering.
123:
1106 l1l7.
Dietrich.
W.E.. 1982.
Settling velocity
of
natural particles. IVater
Re.sourccs
Research.
18:
1615-1626.
DIN (Deutsche Ingenieure Normung),
1983.
DIN18I23: Subsoil:
testing
of
soil samples: determination
of
the particle size distribu-
tion.
Ferguson.
R.,
and Paola.
C.
1997. Bias and precision
of
percentiles
of
bulk grain size distributions. Ftirth
Surface Proees.ses
and
Land-
forms.
22:
1061-1077.
Fieller. N.R.J., Flenley,
E.C., and
Olbrichi,
W.. 1992.
Statistics
of
particle size data.
Applied
Statistics.
41:
127-146.
Folk.
R.L.. and
Ward.
W.C. 1957.
Brazos river
bar: a
study
in the
significance
of
grain size parameters. Journal of
Sedimentary
Pel rol-
ogv.
11:
3-26.
Fripp.
J.B..
and
Diplas.
P.. 1993.
Surface sampling
in
gravel strearns.
Journalof Hydraulic
Engineering.
119: 473-490.
Griffiths. J.C"..
1967.
.Scientific Mettiod
in
.inalysis
of
Sediments.
McGraw-Hill.
Hallermeier.
R.J.. 1981.
Terminal setiling velocity
of
commonly
occurring sand grains.
Sedimentology.
28:
S59-865.
Hey,
R.D.,
and Thorne.
C.R..
1983. Accuracy
of
surface samples frotn
gravel
bed
material. American Sociely
of
Civil Engineers.
Proceed-
ings.
Journal of the Hydraulics
Divi.sion.
109:
842-851.
Ibbeken.
H..
1983. Jointed source rock
and
fluvial gravels controlled
by Rosin's Law:
a
grain-size study
in
Calabria, south Italy. Journal
of Sedimentary Petrology.
53:
(213-1231.
Inman.
D.L..
1949. Sorting
of
sedimenls
in the
light offluid mechanics.
Journal
of
Sedimentary
Petrology.
19:
51 70.
Inman.
D.L.. 1952.
Measures
for
describing
the
size distribution
of
sediments. JouruutolSedinti-ntarv
Pt'trologv.
22: 125- 145,
ISO (International Standards Organization).
1977.
ISO4364-I977.
Liquid llow measurement
in
open channels
bed
material
sam-
pling.
ISO (International Standards Organization), 9l95-i992. Sampling
and
atialysis
of
gravel
bed
tnateiial.
Kellerhals.
R.. and
Bray.
D.I.,
1971. Sampling procedures
for
coarse
Huvial sediments. American Society
of
Civil F.ngincers, Journalof
the Hydraulics Division
97:
1165
-1179.
Kellerhals.
R..
Shaw.
J., and
Arora,
V.K.. 1975. On
grain size frotn
thin sections. Journal of
Geology.
83:
79-96.
Kittieman,
L.R. Jr..
1964. Application
of
Rosin's distribution
in
size-
frequency analysis
of
clastic rocks. Journal
of
Sedimentary
Petrol-
ogy.
34:
483-502.2
Komar.
P.D.. and Cui. B.. 1984. The
analysis
of
grain size
measurements
by
settlinc tube techniques. Journal of Sedimentary
Petrology.
54: 603 614. ^
Krumbein.
W.C 1938.
Size-frequency distributions
and the
normal
phi-curve.
Journal of Sedimentary
Petrology.
8: 84 90.
Krutiibein,
W.C. 1941.
Measuretnent
and
geological significance
of
shape and roundness
of
sedimentary particles. Journal oj
Sedimen-
tary
Petrology.
11:
64 72.
Logan.
B.E.. and
Kilps.
J.R.. 1995.
Fractal geomelry
of
particle
aggregates formed
in
different fluid mechanical environments.
Water
Re.u-areh.
29: 443 453.
Middleton.
G.V., 1970.
Generation
of the
log-normal frequency
distribution
in
sediments.
In
Romanova.
M.A.. and
Sarmonova,
O.V. (eds.).
Topics
in
Mathemaiieal
Geology.
N.Y.:
Consultants
Burcati:
pp.34
42.
[Originally ptiblished
in
Russian.
1968 as the
Andrew Vistelius Anniversary Volume]
Milan.
D.J..
Heritage. G.L.. Larg'e. A.R.G.. and Brunsdon. C.F..
1999.
Inlluence
of
particle shape
and
sorting upon satnple size estimates
for
a
coarse-grained upland stream.
Sedimentary
Geology.
129:
85
100.