#### Transcript Chap2. ppt

```Chapter 2. Grain Texture
Clastic sediment and sedimentary rocks are made up of discrete particles.
The texture of a sediment refers to the group of properties that describe
the individual and bulk characteristics of the particles making up a
sediment:
Individual
Grain Size
Bulk (Grain Size Distribution)
Grain Shape
Grain Orientation
Porosity
}
Permeability
Secondary properties
that are related to the
others.
These properties collectively make up the texture of a sediment or
sedimentary rock.
Each can be used to infer something of:
The history of a sediment.
The processes that acted during transport and deposition of a
sediment.
The behavior of a sediment.
This section focuses on each of these properties, including:
Methods of determining the properties.
The terminology used to describe the properties.
The significance of the properties.
Grain Size
I. Grain Volume (V)
a) Based on the weight of the particle:
m  Vr S
Where:m is the mass of the particle.
V is the volume of the particle.
rs is the density of the material making up the particle.
(r is the lower case Greek letter rho).
1. Weigh the particle to determine m.
2. Determine or assume a density.
(density of quartz = 2650kg/m3)
3. Solve for V.
V 
m
rS
Error due to error in assumed density;
Porous material will have a smaller density and less solid volume so
this method will underestimate the overall volume.
b) Direct measurement by displacement.
b) Direct measurement by displacement.
b) Direct measurement by displacement.
Accuracy depends on how accurately the displaced volume can be
measured.
Not practical for very small grains.
For porous materials this method will underestimate the external volume
of the particle.
c) Based on dimensions of the particle.
Where:d is the diameter of the particle
And the particle is a perfect sphere.
V 
d
6
Measure the diameter of the particle
and solve for V.
Problem: natural particles are rarely spheres.
3
II. Linear dimensions.
a) Direct Measurement
Natural particles normally have irregular shapes so that it is difficult
to determine what linear dimensions should be measured.
Most particles are not spheres so we normally assume that they can be
described as triaxial ellipsoids that are described in terms of three
principle axes:
dL or a-axis
longest dimension.
dI or b-axis
intermediate dimension.
dS or c-axis
shortest dimension.
To define the three dimensions requires a systematic method so that
results by different workers will be consistent.
Sedimentologists normally use the Maximum Tangent Rectangle Method.
Step 1. Determine the plane of maximum projection for the particle.
-an imaginary plane passing through the particle which is in contact with
the largest surface area of the particle.
The maximum projection area is
the area of intersection of the plane
with the particle.
Step 2. Determine the maximum tangent rectangle for the maximum
projection area.
-a rectangle with sides having maximum tangential contact with the
perimeter of the maximum projection area (the outline of the particle)
maximum
tangent
rectangle
Step 2. Determine the maximum tangent rectangle for the maximum
projection area.
-a rectangle with sides having maximum tangential contact with the
perimeter of the maximum projection area (the outline of the particle)
dL is the length of the rectangle.
dI is the width of the rectangle.
Step 3. Rotate the particle so that you view the surface that is at right
angles to the plane of maximum projection.
dS is the longest distance through the particle in the direction normal to
the plane of maximum projection.
The volume of a triaxial ellipsoid is given by:
 d xd xd
L
I
S
V=
6
For fine particles only dL and dI can be measured in thin sections.
Thin sections are 30 micron (30/1000 mm) thick slices of rock through
which light can be transmitted.
http://faculty.gg.uwyo.edu/heller/Sed%20Strat%20Class/SedStratL1/thin_section_mov.htm
Axes lengths measured in thin section are “apparent dimensions” of the
particle.
The length measured in thin
section depends on where in the
particle that the plane of the thin
section passes.
Axes lengths measured in thin section are “apparent dimensions” of the
particle.
The length measured in thin
section depends on where in the
particle that the plane of the thin
section passes.
Axes lengths measured in thin section are “apparent dimensions” of the
particle.
The length measured in thin
section depends on where in the
particle that the plane of the thin
section passes.
For a spherical particle its true
diameter is only seen in thin section
when the plane of the thin section
passes through the centre of the
particle.
The three axes lengths that are commonly measured are often
expressed as a single dimension known as the nominal diameter of a
particle (dn):
dn is the diameter of the sphere with volume (V1) equal the volume (V2)
of the particle with axes lengths dL, dI and dS.
V1 = volume of the sphere.
V2 = volume of a particle.
(a triaxial ellipsoid)
V1 
V2 

d
6

6
3
n
dLdI dS
By the definition of nominal diameter, V1 = V2
Therefore:

6
d 
3
n

6
dLdI ds

d 
6
3
n

6
dLdI ds
dn can be solved by rearranging the terms:
d 
3
n
6



6
dLdI ds

d 
6
3
n

6
dLdI ds
dn can be solved by rearranging the terms:
d 
3
n
6



6
dLdI ds

d 
6
3
n

6
dLdI ds
dn can be solved by rearranging the terms:
d 
3
n
6



6
dLdI ds

d 
3
n
6

6
dLdI ds
dn can be solved by rearranging the terms:
d 
3
n
6



6
dLdI ds
d  dLdI ds
3
n
Therefore:
dn 
3
dLdI ds
Nominal diameter
b) Sieving
Used to determine the grain size distribution
(a bulk property of a sediment).
A sample is passed through a
vertically stacked set of square-holed
screens (sieves).
A set of screens are stacked, largest holes on top, smallest on the bottom
and shaken in a sieve shaker (Rotap shakers are recommended).
Grains that are larger than the holes
remain on a screen and the smaller
grains pass through, collecting on
the screen with holes just smaller
than the grains.
The grains collected on each screen
are weighed to determine the weight
of sediment in a given range of size.
The later section on grain size distributions will explain the method more
clearly.
Details of the sieving method are given in Appendix I of the course notes.
III. Settling Velocity
Another expression of the grain size of a sediment is the settling velocity
of the particles.
Settling velocity (w; the lower case Greek letter omega ): the terminal
velocity at which a particles falls through a vertical column of still water.
Possibly a particularly meaningful expression of grain size as many
sediments are deposited from water.
When a particle is dropped into a column of fluid it immediately
accelerates to some velocity and continues falling through the fluid at
that velocity (often termed the terminal settling velocity).
The speed of the terminal settling velocity of a particle depends on
properties of both the fluid and the particle:
Properties of the particle include:
The size if the particle (d).
The density of the material making up the particle (rs).
The shape of the particle.
a) Direct measurement
Settling velocity can be measured using settling tubes: a
transparent tube filled with still water.
In a very simple settling tube:
A particle is allowed to fall from the top of a column of
fluid, starting at time t1.
The particle accelerates to its terminal velocity
and falls over a vertical distance, L, arriving there
at a later time, t2.
The settling velocity can be determined:
w 
L
t 2  t1
A variety of settling tubes have been devised with different means of
determining the rate at which particles fall. Some apply to individual
particles while others use bulk samples.
i) Tube length: the tube must be long enough so
that the length over which the particle initially
accelerates is small compared to the total length
over which the terminal velocity is measured.
Otherwise, settling velocity will be underestimated.
ii) Tube diameter: the diameter of the tube must be at least 5 times the
diameter of the largest particle that will be passed through the tube.
If the tube is too narrow the particle will be slowed as it settles by the
walls of the tube (due to viscous resistance along the wall).
iii) In the case of tubes designed to measure bulk samples, sample size
must be small enough so that the sample doesn’t settle as a mass of
sediment rather than as discrete particles.
Large samples also cause the risk of developing turbulence in the
column of fluid which will affect the measured settling velocity.
b) Estimating settling velocity based on particle dimensions.
Settling velocity can be calculated using a wide variety of formulae
that have been developed theoretically and/or experimentally.
Stoke’s Law of Settling is a very simple formula to calculate the settling
velocity of a sphere of known density, passing through a still fluid.
Stoke’s Law is based on a simple balance of forces that act on a particle
as it falls through a fluid.
FG, the force of gravity acting to make the
particle settle downward through the fluid.
FB, the buoyant force which opposes the
gravity force, acting upwards.
FD, the “drag force” or “viscous force”,
the fluid’s resistance to the particles
passage through the fluid; also acting
upwards.
Force (F) = mass (m) X acceleration (A)
FG depends on the volume and density (rs) of the particle and is given by:
FG 

d  rs  g 
3

r s gd
3
6
6
FB is equal to the weight of fluid that is displaced by the particle:
FB 

d rg 
3

r gd
3
Where r is the density of the fluid.
6
6
FD is known experimentally to vary with the size of the particle, the
viscosity of the fluid and the speed at which the particle is traveling
through the fluid.
Viscosity is a measure of the fluid’s “resistance” to deformation as the
particle passes through it.
F D  3 d  U
Where  (the lower case Greek letter mu) is the fluid’s dynamic
viscosity and U is the velocity of the particle; 3d is proportional to the
area of the particle’s surface over which viscous resistance acts.
FG 

6
r S gd
FB 
3

r gd
F D  3 d  U
3
6
FG and FB are commonly combined to form the expression for the
“submerged weight” ( FG' ) of the particle; the gravity force less the
buoyant force:
'
FG  FG  F B


6

r S gd 
3

6
FG 
'

r gd
6
 r S  r  gd

6
 rS
3
3
 r  gd
3
“submerged
weight” of the
particle.
The net gravity force acting on a particle falling through the fluid.
We now have two forces acting on the falling particle.
F 
'
G

6
 rS
 r  gd
3
and
Acting downward,
causing the particle to
settle.
F D  3 d  U
Acting upward,
retarding the settling of
the particle.
What is the relationship between these two forces at the terminal settling
velocity?
'
They are equal: FD = FG
Stoke’s Law is based on this balance of forces.
FD  F
'
G
Where F D  3 d  U and F 
'
G
Such that:
3 d  U 

6
r S

6
 rS
 r  gd
 r  gd
3
3
The settling velocity can be determined by solving for U, the velocity of
the particle, so that U = w. Therefore:
3 d w 

6
 r S  r  gd
3
3 d w 

6
 r S  r  gd
3
Rearranging the terms:
w 

6
 rS
w 
2
3
 r  gd 
 r S  r  gd
1
3 d 
2
18 
Stoke’s Law of Settling
Example:
A spherical quartz particle with a diameter of 0.1 mm falling through
still, distilled water at 20C
d = 0.0001m
rs= 2650kg/m3
 = 1.005  10-3 Ns/m2
g = 9.806 m/s2
w 
 r S  r  gd
r = 998.2kg/m3
2
18 
Under these conditions (i.e., with the values listed above) Stoke’s Law
reduces to:
w   8.954  10
5

d
2
For a 0.0001 m particle: w = 8.954  103 m/s or  9 mm/s
Stoke’s Law has several limitations:
i) It applies well only to perfect spheres (in deriving Stoke’s Law the
volume of spheres was used).
The drag force (3dw) is derived experimentally only for spheres.
Non-spherical particles will experience a different distribution of viscous
drag.
ii) It applies only to still water.
Settling through turbulent waters will alter the rate at which a particle
settles; upward-directed turbulence will decrease w whereas downwarddirected turbulence will increase w.
iii) It applies to particles 0.1 mm or finer.
iii) It applies to particles 0.1 mm or finer.
Coarser particles, with larger settling velocities, experience different
forms of drag forces.
Stoke’s Law overestimates
the settling velocity of quartz
density particles larger than
0.1 mm.
When settling velocity is low
(d<0.1mm) flow around the
particle as it falls smoothly
follows the form of the sphere.
When settling velocity is high
(d>0.1mm) flow separates from
the sphere and a wake of eddies
develops in its lee.
Drag forces (FD) are only due to the
viscosity of the fluid.
Pressure forces acting on the
sphere vary.
Negative pressure in the
lee retards the passage
new resisting force.
Stoke’s Law neglects
resistance due to
pressure.
iv) Settling velocity is temperature dependant because fluid viscosity and
density vary with temperature.
Temp.
C

Ns/m2
r
w
Kg/m3 mm/s
0
1.792  10-3
999.9
5
100
2.84  10-4
958.4
30
Grain size is sometimes described as a linear dimension based on Stoke’s
Law:
Stoke’s Diameter (dS): the diameter of a sphere with a Stoke’s settling
velocity equal to that of the particle.
w 
r S
 r  gd
2
18 
Set d = dS and solve for dS.
dS 
18 w
r S
 r g
Grade scales define limits to a range of grain sizes for a given class
Sedimentologists use the Udden-Wentworth Grade Scale.
Sets most boundaries to
vary by a factor of 2.
They provide a basis for
a terminology that
describes grain size.
e.g., medium sand falls
between 0.25 and 0.5 mm.
Sedmentologists often express grain size in units call Phi Units (f; the
lower case Greek letter phi).
Phi units assign whole numbers to the boundaries between size classes.
Phi was originally defined as:
f   log
To make Phi dimensionless it was
later defined as:
f   log
d ( mm )
2
dO
Where dO = 1 mm.
2
d ( mm )
Phi is the negative of the power to which 2 is raised such that it
equals the dimension in millimetres.
f   log 2 1( mm )  0f
2-0 = 1
f   log 2 .25( m m )  2f
2-2 = 0.25
f   log 2 0.125( m m )  3f
2-3 = 0.125
f   log 2 64( m m )   6f
2-(-6) = 64
With a calculator you can convert
millimetres to Phi:
 log 10 d ( mm ) 

f   

log
2
10


You can convert Phi to millimetres:
d (mm )  2
f
Note that when grain size is plotted as phi units grain size becomes
smaller towards the right.
V. Describing Grain Size Distributions
a) Grain Size Data
Data on grain size distributions are normally collected by sieving.
1. Grain Size
Class (f)
2. Weight
(grams)
3. Weight
(%)
4. Cumulative Weight (%)
-0.5
0.40
1.3
1.3
0
1.42
4.6
5.9
0.5
2.76
8.9
14.8
1.0
4.92
15.9
30.7
1.5
5.96
19.3
50.0
2.0
5.96
19.3
69.3
2.5
4.92
15.9
85.2
3.0
2.76
8.9
94.1
3.5
1.42
4.6
98.7
4.0
0.40
1.3
100
Total:
30.92
100
1. Grain size class: the size of holes on which the weighed sediment
was trapped in a stack of sieves.
2. Weight (grams): the weight, in grams, of sediment trapped on the
sieve denoted by the grain size class.
3. Weight (%): the weight
of sediment trapped
expressed as a percentage of
the weight of the total
sample.
4. Cumulative Weight (%):
the sum of the weights
expressed as a percentage
(column 3).
1. Grain Size Class
(f)
2. Weight
(grams)
3. Weight
(%)
4. Cumulative
Weight (%)
-0.5
0.40
1.3
1.3
0
1.42
4.6
5.9
0.5
2.76
8.9
14.8
1.0
4.92
15.9
30.7
1.5
5.96
19.3
50
2.0
5.96
19.3
69.3
2.5
4.92
15.9
85.2
3.0
2.76
8.9
94.1
3.5
1.42
4.6
98.7
4.0
0.40
1.3
100
Total:
30.92
100
Each value in column 4 is the percentage of the sample that is coarser
than the screen on which the sediment was trapped.
1. Grain Size Class
(f)
2. Weight
(grams)
3. Weight
(%)
4. Cumulative
Weight (%)
-0.5
0.40
1.3
1.3
0
1.42
4.6
5.9
0.5
2.76
8.9
14.8
1.0
4.92
15.9
30.7
1.5
5.96
19.3
50
2.0
5.96
19.3
69.3
2.5
4.92
15.9
85.2
3.0
2.76
8.9
94.1
3.5
1.42
4.6
98.7
4.0
0.40
1.3
100
Total:
30.92
100
Each value in column 4 is the percentage of the sample that is coarser
than the screen on which the sediment was trapped.
30.7% of the total sample is
coarser than 1.0 f.
1. Grain Size Class
(f)
2. Weight
(grams)
3. Weight
(%)
4. Cumulative
Weight (%)
-0.5
0.40
1.3
1.3
0
1.42
4.6
5.9
0.5
2.76
8.9
14.8
1.0
4.92
15.9
30.7
1.5
5.96
19.3
50
2.0
5.96
19.3
69.3
2.5
4.92
15.9
85.2
3.0
2.76
8.9
94.1
3.5
1.42
4.6
98.7
4.0
0.40
1.3
100
Total:
30.92
100
Each value in column 4 is the percentage of the sample that is coarser
than the screen on which the sediment was trapped.
30.7% of the total sample is
coarser than 1.0 f.
85.2% of the total sample is
coarser than 2.5 f.
1. Grain Size Class
(f)
2. Weight
(grams)
3. Weight
(%)
4. Cumulative
Weight (%)
-0.5
0.40
1.3
1.3
0
1.42
4.6
5.9
0.5
2.76
8.9
14.8
1.0
4.92
15.9
30.7
1.5
5.96
19.3
50
2.0
5.96
19.3
69.3
2.5
4.92
15.9
85.2
3.0
2.76
8.9
94.1
3.5
1.42
4.6
98.7
4.0
0.40
1.3
100
Total:
30.92
100
b) Displaying Grain Size Data
i) Histograms
Readily shows the relative amount of sediment in each size class.
Each bar width equals
the class interval (0.5 f
intervals in this case).
Bars extend from the
maximum size to the
minimum size for each
size class.
ii) Frequency Curves
A smooth curve that joins the midpoints of each bar on the histogram.
iii) Cumulative Frequency Curves
A smooth curve that represents the size distribution of the sample.
Several curves for different
samples can be plotted together on
one diagram for comparison of the
samples.
Sedimentologists commonly plot cumulative frequency curves on a
probability scale for the cumulative frequency.
On such plots normal, bell shaped distributions plot as a straight line.
Plots of samples which are made up of normally distributed
subpopulations plot as a series of straight line segments, each segment
representing a normally distributed subpopulation.
Plots of samples which are made up of normally distributed
subpopulations plot as a series of straight line segments, each segment
representing a normally distributed subpopulation.
A benefit of cumulative frequency plots is that percentiles can be taken
direction from the graph.
fn is the grain size that is finer than n% of the total sample.
fn is referred to as the nth
percentile of the sample.
In the example f20 is 0.86f
0.86f is that grain size that is
finer than 20% of the sample.
Conversely, 0.86f is coarser
than 80% of the sample.
c) Describing Grain Size Distributions.
Folk and Ward (1957) introduced the Graphic Method to estimate the
various statistical parameters describing a grain size distribution using
only percentiles taken from cumulative frequency curves.
Median
M d  f 50
Mean
M 
3
Standard deviation
 
Skewness
Sk 
Kurtosis
f16  f 50  f 84
f 84  f 16

f 95  f 5
4
K 
6 .6
f 84  f16  2f 50
2 f 84  f 16 
f 95  f 5
2 . 44 (f 75  f 25 )

f 95  f 5  2f 50
2 f 95  f 5 
i) Median (Md)
The midpoint of the distribution; the 50th percentile.
50% of the sample is finer than the median and 50% of the sample is
coarser than the median.
In this example the Median is
approximately 0.35f.
ii) Mean (M)
The arithmetic average of the distribution.
If the distribution is purely symmetrical M = Md.
The Udden-Wentworth Scale is used to define terms to describe the
sediment based on the mean size.
Eg., if M = 0.34f, the
sample is said to be
coarse sand.
Example calculation of the Mean:
Mean
M 
f16  f 50  f 84
3
1. Determine f16, f50and f84
Example calculation of the Mean:
Mean
M 
f16  f 50  f 84
3
1. Determine f16, f50and f84
f16 = -0.59f
Example calculation of the Mean:
Mean
M 
f16  f 50  f 84
3
1. Determine f16, f50and f84
f16 = -0.59f
f50 = 0.35f
Example calculation of the Mean:
Mean
M 
f16  f 50  f 84
3
1. Determine f16, f50and f84
f16 = -0.59f
f50 = 0.35f
f84 = 1.27f
Example calculation of the Mean:
Mean
M 
f16  f 50  f 84
3
1. Determine f16, f50and f84
f16 = -0.59f
f50 = 0.35f
f84 = 1.27f
M 
 0.59  0.35  1.27
3
= 0.34f
iii) Standard Deviation (; lower case Greek letter sigma)
Also referred to as the sorting coefficient or dispersion coefficient of a
sediment.
The units of sorting are phi units.
A measure of how much variation in grain size is present within a sample.
68% of the total weight of
sediment in a sample falls within
+/- 1 standard deviation of the
mean.
E.g., M = 0.34f and  = 0.75f
68% of the sample falls in the range from -0.41 to 1.09f.
Sedimentologists use a specific terminology to describe the sorting of a
sediment:
Very well sorted
0 <  < 0.35f
Well sorted
0.35 <  < 0.50f
Moderately well sorted
0.5 <  < 0.71f
Moderately sorted
0.71 <  < 1.00f
Poorly sorted
1.00 <  < 2.00f
Very poorly sorted
2.00 <  < 4.00f
Extremely poorly sorted
 > 4.00f
Examples of sorting:
iv) Skewness (Sk)
A measure of the symmetry of the distribution.
Values range from –1.0 to +1.0.
Sk = 0
Perfectly symmetrical.
Md = M
Sk > 0
The distribution has more fine particles than a symmetrical distribution
would have.
The distribution is said to be fine tailed.
M is finer than Md
Sk < 0
The distribution has more coarse particles than a symmetrical
distribution would have.
The distribution is said to be coarse tailed.
M is coarser than Md
Terminology:
Sk > 0.3 strongly fine skewed
0.1 < Sk < 0.3 fine skewed
-0.1 < Sk < 0.1 near symmetrical
-0.3 < Sk < -0.1 coarse skewed
Sk < -0.3 strongly coarse skewed
v) Kurtosis (K)
A measure of the peakedness of the distribution (related to sorting).
K > 1 sharp peaked (Leptokurtic)
K = 1 normal (Mesokurtic)
K < 1 flat peaked (Platykurtic)
VI. Implications of Grain Size
Grain size is a fundamental property of any granular material.
It influences other fundamental properties.
Historically it was hoped that ancient depositional environments could be
determined on the basis of grain size and grain size distributions.
When drilling wells (oil, gas, water) the most abundant samples are
small pieces of rock called “drill chips”.
Anatomy of a Rotary Drilling Rig
Drill bit:
Boron alloy buttons +/- diamond grit.
Anatomy of a Rotary Drilling Rig
Mud is pumped through the drill
string to the bit; as the mud rises
to the surface it carries “drill
chips” along with it.
Drill Chips:
1 to 5 millimetre diameter pieces of rock.
Collected and bagged as the mud brings
the chips to the surface.
1 sample bag represents 3 metres of
drilled rock.
One attempt to distinguish depositional environments on the basis of
grain size distribution focused on beach versus river sands.
Samples were collected from rivers and beaches (both lake and ocean
beaches) and Skewness was plotted against Sorting Coefficient.
Beaches tend to have sands that are
better sorted and with more common
coarse tail skewness than river
sands.
This reflects the difference in
processes that act in rivers and
beaches.
Rivers transport a wide range of
grain sizes: large particles move in
contact with the bed and large
volumes of fine particles are carried
in suspension in the current.
Their deposits tend to be relatively poorly sorted and rich in fine particles
(+ve or fine tailed skewness).
Beaches experience repeated swash and backwash of waves running up
the beach face.
The repeated action of the currents washes fine sand from the beach
(improving its sorting) and leaving larger grains behind (causing coarse
tail skewness).
The difference in processes acting on beaches and in rivers results in
distinct differences in their grain size distributions.
Other attempts to apply this method had limited results.
The problem is that grain size distribution can be inherited from the
source material that makes up the sediment.
If a beach forms on ancient river sediments then the beach deposits will
inherit characteristics of river deposits.
If a river erodes through ancient beach deposits then its sediment will
bear the characteristics of beach sediments.
In rivers it has been found that, overall, sorting of a sediment improves
with the distance of transport.
The longer the distance of transport the greater the opportunity to
remove fine material (in suspension) and to leave coarser material
behind, reducing the range of grain sizes present (improving the sorting
and decreasing the sorting coefficient).
From: Gomez, Rosser, Peacock, Hicks and Palmer, 2001, Downstream fining in a rapidly aggrading gravel
bed river. Water Resources Research, v. 37, p. 1813-1823.
Why measure Grain Size?
1. It is a fundamental property of any granular material.
2. It influences a variety of other properties.
3. It gives clues to the history of a sediment (more details later in the
course).
Grain Shape
An individual property (rather than a bulk property) that is as
fundamental as grain size.
Shape can be described in a variety of ways:
Roundness: a description of how angular the edges of a particle are.
Sphericity: how closely a particle’s shape resembles a sphere.
Form: the overall appearance of a particle.
Surface texture: scratches, pits, grooves, etc. on a particle’s surface.
I. Roundness
Several methods of description; some more practical than others.
Time consuming and very impractical but results are reliable.
RW: the ratio of the average radius of curvature of the corners on the
surface of a grain to the radius of curvature of the largest circle that can
be inscribed within the projection of the particle.
RW 
r
N
1
R

r
NR
Where N is the number of corners.
RW approaches 1 for a perfectly round particle.
b) Dobson and Fork Roundness (RF)
RF: the ratio of the radius of curvature of a particle’s sharpest corner (r)
to the radius of curvature of the largest circle that can be inscribed on
the projection of the particle (R).
RF 
c) Power’s visual comparison chart.
r
R
The most commonly used method of determining roundness.
Defines terms for describing roundness.
e.g., 0.25 < RW <0.35 is termed sub-angular.
Also gives a “shape parameter” (r) which is a logarithmic transformation
of RW so that the boundaries between roundness classes are whole
numbers.
II. Sphericity
A measure of how closely a particle resembles a sphere.
Usually denoted by Y, the capital Greek letter psi.
A measure of sphericity is useful to determine whether or not Stoke’s
Law of settling is applicable (e.g., how much a particle differs from a
sphere).
Sphericity determines the use of a sediment (e.g., high sphericity
sediment is not particularly good for making concrete).
Y: the ratio of the diameter of a sphere with the same volume as the
particle to the volume diameter of a sphere that will circumscribe the
particle.
Y 
3
VS
VC
Where VS is the volume of the particle and VC is the
volume of the circumscribing sphere.
There are a couple of methods of determining Y.
Y 
3
VS
VC
Where VS is the volume of the particle and VC is the
volume of the circumscribing sphere.
i) Based on volume measurement.
Step 1. Measure the volume of the particle by displacement in water in
a graduated cylinder. This determines VS.
Step 2. Measure the long axis of the particle (dL) as this will be the
diameter of the largest circumscribing sphere.
VC 

6
By this procedure:
Y 
3
VS 
d
3
L

6
d
3
L
Y 
3
Where VS is the volume of the particle and VC is the
volume of the circumscribing sphere.
VS
VC
ii) Approximate VS by assuming that the particle is a triaxial ellipsoid.
VS 

6
Y 
VC 
and
dLdIdS

3
6
dLdI dS 
Y 
3

6
3
dL
6
d
32
L
dIdS
d
2
L
By either method, as Y approaches 1 the particle approaches a sphere in
overall shape (i.e., for a perfect sphere Y = 1).
b) Sneed and Folk (YP) or Maximum Projection Sphericity
YP: the ratio of the maximum projection area of a sphere with volume
equal to the particle to the maximum projection area of the particle.
2
YP 
3
dS
dLdI
Sneed and Folk argued that it was the projection area of a particle that
experienced the viscous drag of moving fluid, therefore it was more
important than the volume of the particle.
Y 1
With increasing sphericity.
c) Corey Shape Factor (SF)
Not really a measure of sphericity but similar to YP.
S .F . 
dS
dLdI
Used in a variety formulae used to calculate sediment transport rates.
d) Riley Sphericity (YR)
Used when only a two dimensional view
is available on thin sections.
YR 
di
dc
III. Form
Provides a consistent terminology for describing the overall form of
particles.
Based on various ratios of dL, dI and dS
a) Zingg Form Index
Assigns specific shape terms based
on ratios of dI/dL and dS/dI.
Independent of sphericity although
equant particles are highly
spherical.
b) Sneed and Folk Form Index
Defines 10 shape classes.
Shows sphericity for each class.
IV. Implications of particle shape
a) Controls on particle shape.
i) Lithology
For particles that are rock fragments aspects of the lithology of the
source rock can influence shape.
Massive source rocks (non-bedded) tend to produce
more equant particles (e.g., granite, massive
sandstone, etc.).
Bedded and foliated rocks tend to produce
more platy particles (e.g., well-bedded
limestones).
ii) Hardness
Particularly hard clasts (e.g., granite, quartz-cemented sandstone, etc.)
change in shape during transport less readily than softer lithologies.
Softer lithologies (e.g., limestone) change shape much more readily.
Unconsolidated material (e.g., cohesive mud) changes shape almost
immediately when it is transported.
b) Changes in Shape by Transport
Transport of particles by water, wind or flowing glaciers has the potential
to cause changes to their shape over time.
During transport particles interact with each other and with the surface
over which they move.
Shape is modified by grinding, chipping and crushing that takes place
due to this interaction.
The following figure reports data derived from long distance transport of
blocks cut initially as cubes within a circular flume.
Changes in roundness and sphericity for cubes of chert (a very hard rock
type) and softer limestone.
Roundness increases for limestone much more quickly with transport
than the harder chert, especially during the early phase of transport.
The rate of increase in roundness is greatest during early transport for
both rock types.
As rounding takes place it becomes increasingly more difficult to change
the shape to further improve roundness.
With an initially angular particle it is relatively easy to increase
roundness by breaking off the sharp corners.
As the particle becomes rounder, larger and larger masses of material
must be removed to cause a comparable increase in roundness.
Sphericity changes shape only very slowly because the particles began as
relatively equant shaped cubes (i.e., with high sphericity to begin with).
Relatively large masses of material must be removed to significantly
change the sphericity of a particle.
Gray area must be
removed to increase
sphericity.
Gravel size material changes shape relatively rapidly with transport in
comparison to sand size material.
Shape change in sand can be inferred from rates of change of weight of
particles with transport; changes in shape require changes in mass.
For quartz grains in the size range 2 mm to 0.05 mm, that are transported
in water, the weight loss is <0.1% per 1000 km of transport (the rate is
doubled for feldspars).
For quartz grains finer than 0.05 mm that are transported in water there is
virtually no change in weight with transport.
Why so little change in very fine sand?
Breakage takes place during collisions (between grains and a solid
substrate) and depends on the amount of momentum that is exchanged
during collisions.
Momentum = mass  velocity
Very fine sand, silt and clay particles have very small mass so that they
have little momentum.
During collisions there is little momentum exchanged and, therefore,
little breakage which is required to change the mass and shape of the
grains.
Momentum is further reduced when transport is in water because the
buoyant force reduces its effective weight.
Most of the particle’s momentum is used in “pushing” the water as the
particle moves, losing momentum due to the viscosity of the water.
When transport is in water there is little momentum to cause breakage
and change in shape.
Finally, we’ll see later that particles finer than 0.05 mm are transported
in suspension (floating in the water column), further reducing the
chances of shape-changing collisions.
Transport by wind is more effective in changing the shape of fine grains.
Air has low density (little buoyant force) and very low viscosity so that
there is more momentum exchanged during collisions.
Rate of weight loss is 100 to 1000 times that when transport is in water.
Eolian desert sands tend to have relatively high sphericity.
Shape Sorting by Transport
The roundness and sphericity of particles influence the ease with
flowing water can move the particle.
To cause an angular cube to roll the fluid
force acting on it must overcome the weight
of the cube and pivot it 45 before its centre
of mass passes over the pivot point.
Once the centre of mass passes over the
pivot point the weight of the particle aids
in the motion.
A rounder, octagonal particle need only be
pivoted over 20 before it’s center of mass
passes over the pivot point and the weight
contributes to the motion.
For a spherical particle the angle that must be
exceeded is very close to 0.
Overall, the more spherical the particle the more
readily it will transported by a moving fluid.
“Shape sorting” involves the selective removal of particles due to their
shape.
Time 1: angular and spherical particles are introduced to a current.
Time 2: the spheres are preferentially removed as they roll away from
the site due to the current. The remaining particles are largely angular
and the particles down stream are more rounded and spherical.
b) Depositional environments and particle shape
Roundness and sphericity may be useful in helping determine ancient
environments or processes associated with deposition.
Glacial till that is transported within glacial ice is typically angular in
shape.
Angularity reflects a lack of transport by water prior to deposition.
Eolian (windblown) sands commonly have a higher sphericity than
water transported sand.
In rivers changes in roundness and sphericity with transport are well
documented.
Demir, 2003, Downstream changes in bed material size and shape characteristics in a small upland stream:
Cwm Treweryn, in South Wales, Yerbilimleri, v. 28, p. 33-47.
It has been found that, in
general, river gravels are
relatively compact whereas
beach gravels tend to be
more platy or disc-shaped.
As for grain size distribution, it is the processes in the two environments
that differ and lead to the characteristic shapes.
In rivers, gravel rolls along the bottom so that more equant or spherical
shapes are most commonly transported.
On beach faces the swash and backwash may play two rolls in enhancing
the enrichment by disc-shaped particles:
i) By selectively removing more spherical particles which readily roll
down the slope of the beach face, leaving flatter particles behind.
ii) The swash and backwash may lead to back and forth motion of the
particle on the beach face.
This leads to abrasion of one side and if the waves cause it to flip over
abrasion takes place on the other side, ultimately leading to a disc-shaped
clast.
Time 1
Time 2
Time 3
Time 4
Problem: lithology plays an important roll in determining shape.
e.g., a river with a well-bedded limestone source of gravel will have
predominantly platy gravel.
Furthermore, like grain size distribution, shape may be inherited from the
source material of the particles.
A river that erodes through ancient beach gravel will have clasts that are
platy in form.
http://www.sandcollectors.org
Porosity and Permeabilty
Both are important properties that are related to fluids in sediment and
sedimentary rocks.
Fluids can include: water, hydrocarbons, spilled contaminants.
Most aquifers are in sediment or sedimentary rocks.
Virtually all hydrocarbons are contained in sedimentary rocks.
Porosity: the volume of void space (available to contain fluid or air) in a
sediment or sedimentary rock.
Permeability: related to how easily a fluid will pass through any
granular material.
I. Porosity (P)
The proportion of any material that is void space, expressed as a
percentage of the total volume of material.
P 
VP
Where VP is the total volume of pore space
and VT is the total volume of rock or
sediment.
 100
VT
In practice, porosity is commonly based on measurement of the total
grain volume of a granular material:
P 
VT  V G
VT
 100
Where VG is the total volume of
grains within the total volume of
rock or sediment.
 V P  VT  V G
Porosity varies from 0% to 70% in natural sediments but exceeds 70%
for freshly deposited mud.
Several factors control porosity.
a) Packing Density
Packing density: the arrangement of the particles in the deposit.
The more densely packed the particles the lower the porosity.
e.g., perfect spheres of uniform size.
Porosity varies from 0% to 70% in natural sediments but exceeds 70%
for freshly deposited mud.
Several factors control porosity.
a) Packing Density
Packing density: the arrangement of the particles in the deposit.
The more densely packed the particles the lower the porosity.
e.g., perfect spheres of uniform size.
Porosity can vary
from 48% to 26%.
Shape has an important effect on packing.
Tabular rectangular particles can vary from 0% to just under 50%:
Natural particles such as shells can have very high porosity:
In general, the greater the angularity of the particles the more open the
framework (more open fabric) and the greater the possible porosity.
b) Grain Size
On its own, grain size has no influence on porosity!
Consider a cube of sediment of
perfect spheres with cubic
packing.
P 
VT  V G
 100
VT
d = sphere diameter; n = number of grains along a side (5 in this example).
P 
VT  V G
 100
VT
Length of a side of the cube = d  n = dn
Volume of the cube (VT):
VT  dn  dn  dn  d n
3
3
Total number of grains: n  n  n = n3
Volume of a single grain:
V 

6
Total volume of grains (VG):
VG  n 
3

6
d n
3
3

6
d
3
d
3
P 
VT  V G
 100
VT
Where: VT  d n
3
d n n
3
Therefore:
3
Therefore:

6
P
3
d n
3
d
and
VG  n
3

d
6
3
 100
 

d n 1  
6

P 
 100
3 3
d n
3
Rearranging:
3
3
3
 

P   1    100  48%
6

d (grain size) does not affect the porosity so that porosity is independent
of grains size.
No matter how large or small the spherical grains in cubic packing have
a porosity is 48%.
3
There are some indirect relationships between size and porosity.
i) Large grains have higher settling velocities than small grains.
When grains settle through a fluid the large grains will impact the
substrate with larger momentum, possibly jostling the grains into tighter
packing (therefore with lower porosity).
ii) A shape effect.
Unconsolidated sands tend to
decrease in porosity with
increasing grain size.
Consolidated sands tend to
increase in porosity with
increasing grain size.
Generally, unconsolidated sands undergo little burial and less
compaction than consolidated sands.
Fine sand has slightly higher porosity.
Fine sand tends to be more angular than coarse sand.
Therefore fine sand will support a more open framework (higher
porosity) than better rounded, more spherical, coarse sand.
Consolidated sand (deep burial, well compacted) has undergone
exposure to the pressure of burial (experiences the weight of overlying
sediment).
Fine sand is angular, with sharp edges, and the edges will break under
the load pressure and become more compacted (more tightly packed
with lower porosity).
Coarse sand is better rounded and less prone to breakage under load;
therefore the porosity is higher than that of fine sand.
c) Sorting
In general, the better sorted the sediment the greater the porosity.
In well sorted sands fine grains are not available to fill the pore spaces.
This figure shows the relationship between sorting and porosity for
clay-free sands.
Overall porosity decreases with increasing sorting coefficient (poorer
sorting).
For clay-free sands the reduction in porosity with increasing sorting
coefficient is greater for coarse sand than for fine sand.
The difference is unlikely if clay was also available to fill the pores.
For clay-free sands the silt and fine sand particles are available to fill
the pore space between large grains and reduce porosity.
Because clay is absent less
relatively fine material is not
available to fill the pores of fine
sand.
Therefore the pores of fine sand
will be less well-filled (and have
porosity higher).
d) Post burial changes in porosity.
Includes processes that reduce and increase porosity.
Porosity that develops at the time of deposition is termed primary
porosity.
Porosity that develops after deposition is termed secondary porosity.
Overall, with increasing
burial depth the porosity of
sediment decreases.
50% reduction in porosity
with burial to 6 km depth due
to a variety of processes.
i) Compaction
Particles are forced into closer packing by the weight of overlying
deposits, reducing porosity.
May include breakage of grains.
Most effective if clay minerals are present (e.g., shale).
Freshly deposited mud may have 70% porosity but burial under a
kilometre of sediment reduces porosity to 5 or 10%.
ii) Cementation
Precipitation of new minerals from pore waters causes cementation of
the grains and acts to fill the pore spaces, reducing porosity.
Most common cements are calcite and quartz.
Here’s a movie of
cementation at Paul
Heller’s web site.
iii) Clay formation
Clays may form by the chemical alteration of pre-existing minerals after
burial.
Feldspars are particularly common clay-forming minerals.
Clay minerals are very fine-grained and may accumulate in the pore
spaces, reducing porosity.
Eocene Whitemud
iv) Solution
If pore waters are undersaturated with respect to the minerals making up
a sediment then some volume of mineral material is lost to solution.
Calcite, that makes up limestone, is relatively soluble and void spaces
that are produced by solution range from the size of individual grains to
caverns.
Quartz is relatively soluble when pore waters have a low Ph.
Solution of grains reduces VG, increasing porosity.
Solution is the most effective means of creating secondary porosity.
v) Pressure solution
The solubility of mineral grains increases under an applied stress (such
as burial load) and the process of solution under stress is termed
Pressure Solution.
The solution takes place at the grain contacts where the applied stress is
greatest.
Pressure solution results in a reduction in porosity in two different ways:
1. It shortens the pore spaces as the grains are dissolved.
2. Insoluble material within the grains accumulates in the pore spaces as
the grains are dissolve.
v) Fracturing
Fracturing of existing rocks creates a small increase in porosity.
Fracturing is particularly important in producing porosity in rocks with
low primary porosity.
Why is porosity important?
Especially because it allows us to make estimations of the amount of
fluid that can be contained in a rock (water, oil, spilled contaminants,
etc.).
Example from oil and gas exploration:
Why is porosity important?
Especially because it allows us to make estimations of the amount of
fluid that can be contained in a rock (water, oil, spilled contaminants,
etc.).
Example from oil and gas exploration:
Why is porosity important?
Especially because it allows us to make estimations of the amount of
fluid that can be contained in a rock (water, oil, spilled contaminants,
etc.).
Example from oil and gas exploration:
Why is porosity important?
Especially because it allows us to make estimations of the amount of
fluid that can be contained in a rock (water, oil, spilled contaminants,
etc.).
Example from oil and gas exploration:
Why is porosity important?
Especially because it allows us to make estimations of the amount of
fluid that can be contained in a rock (water, oil, spilled contaminants,
etc.).
Example from oil and gas exploration:
How much oil is contained in the discovered unit?
In this case, assume that the pore
spaces of the sediment in the oilbearing unit are full of oil.
Therefore, the total volume of oil is
the total volume of pore space (VP)
in the oil-bearing unit.
P 
VP
 100
Total volume of oil = VP, therefore solve for VP.
VT
VP 
VT  800 m  200 m  1m  160, 000 m
P  VT
P  10%
100
Therefore:
VP 
10  160, 000
100
 16, 000m
3
of oil
3
II. Permeability (Hydraulic Conductivity; k)
Stated qualitatively: permeability is a measure of how easily a fluid will
flow through any granular material.
More precisely, permeability (k) is
an empirically-derived parameter
in D’Arcy’s Law, a Law that
predicts the discharge of fluid
through a granular material.
Q k
D’Arcy’s Law:
A p
L
Another way to express D’Arcy’s Law is as the flow rate as the
“apparent velocity” (V) of the fluid through the material where:
V 
Q
A
Thus, D’Arcy’s Law can be expressed as:
V 
Q
A
Expanded:
k
A p
L
V  k  p 
1


1
A

1
L
k
p
L
V  k  p 
1


1
L
So, the apparent velocity of a fluid flowing through a granular material
depends on several factors:
p: this is the driving force behind the flow of fluid through granular
materials.
The greater the change in pressure the greater the rate of flow.
(try blowing pop out of a straw!)
V  k  p 
1

:
1


1
L
Apparent velocity decreases with increasing dynamic viscosity.
The higher the viscosity the more difficult it is for the pressure to
“push” the fluid through the small pathways within the material.
(try sucking molasses through a straw!)
1
L
:
The longer the pathway of the fluid the more pressure is needed to
“push” the fluid through the material.
(try drinking a milkshake through a 1 metre long straw!)
This is a viscous effect: resistance to deformation is cumulative
along the length of the tube: the longer the tube (or pathway) the
greater the total resistance.
Those are all properties that are independent of the granular material.
There are also controls on permeability that are exerted by the granular
material and are accounted for in the term (k) for permeability:
k is proportional to all sediment properties that influence the flow of fluid
through any granular material (note that the dimensions of k are cm2).
Two major factors:
1. The diameter of the pathways through which the fluid moves.
2. The tortuosity of the pathways (how complex they are).
1. The diameter of the pathways.
Along the walls of the pathway the velocity is zero (a no slip boundary)
and increases away from the boundaries, reaching a maximum towards
the middle to the pathway.
Narrow pathway: the region where the velocity is low is a relatively
large proportion of the total cross-sectional area and average velocity is
low.
Large pathway: the region where
the velocity is low is proportionally
small and the average velocity is
greater.
It’s easier to push fluid through a large
Pathway than a small one.
2. The tortuosity of the pathways.
Tortuosity is a measure of how
much a pathway deviates from a
straight line.
2. The tortuosity of the pathways.
Tortuosity is a measure of how
much a pathway deviates from a
straight line.
The path that fluid takes through a
granular material is governed by
how individual pore spaces are
connected.
The greater the tortuosity the
lower the permeability because
viscous resistance is cumulative
along the length of the pathway.
Pathway diameter and tortuosity are controlled by the properties of the
sediment and determine the sediment’s permeability.
The units of permeability are Darcies (d):
1 darcy is the permeability that allows a fluid with 1 centipoise
viscosity to flow at a rate of 1 cm/s under a pressure gradient of 1
atm/cm.
Permeability is often very small and expressed in millidarcies (
1
1000
d
)
a) Sediment controls on permeability
i) Packing density
Tightly packed sediment has smaller
pathways than loosely packed
sediment (all other factors being
equal).
Smaller pathways reduce porosity and the size of the pathways so the
more tightly packed the sediment the lower the permeability.
ii) Porosity
In general, permeability increases with primary porosity.
The larger and more abundant the pore spaces the greater the
permeability.
Pore spaces must be well connected
to enhance permeability.
Shale, chalk and vuggy rocks (rocks with large solution holes) may have
very high porosity but the pores are not well linked.
The discontinuous pathways result in low permeability.
Fractures can greatly enhance permeability but do not increase porosity
significantly.
A 0.25 mm fracture will pass fluid
at the rate that would be passed
by13.5 metres of rock with 100 md
permeability.
iii) Grain Size
Unlike porosity, permeability increases with grain size.
The larger the grain size the larger the pore area.
For spherical grains in cubic packing:
Pore area = 0.74d2
A ten-fold increase in grain size yields a hundred-fold increase in
permeability.
iv) Sorting
The better sorted a sediment is the
greater its permeability.
In very well sorted sands the pore
spaces are open.
In poorly sorted sands fine grains
occupy the pore spaces between
coarser grains.
v) Post-burial processes
Like porosity, permeability is changed following burial of a sediment.
In this example permeability
is reduced by two orders of
magnitude with 3 km of
burial.
Cementation
Clay formation
Compaction
Pressure solution
All act to reduce permeability
b) Directional permeability
Permeability is not necessarily isotropic (equal in all directions)
Fractures are commonly aligned in the same direction, greatly
enhancing permeability in the direction that is parallel to the
fractures.
Variation in grain size and geological structure can create directional
permeability.
size becomes finer upwards in
a bed.
Fluid that is introduced at the surface will follow a path that is towards the
direction of dip of the beds.
Fabric (preferred orientation of the grains in a sediment) can cause
directional permeability.
E.g., A sandstone unit of prolate particles.
The direction along the long axes of grains will have larger pathways
and therefore greater permeability than the direction that is parallel to
the long axes.
Grain Orientation
Fabric: the group of properties that are related to the spatial
arrangement of the particles (including packing and orientation).
The term is commonly used to refer to orientation only.
Why is it important?
1. It can affect other properties.
e.g., permeability, how it breaks (building stone).
2. It can have genetic significance.
The problem with grain size and shape was that they may be inherited
from their source rock.
Particle orientation is achieved at the time of deposition in response to
processes that acted in the environmental setting.
It remains fixed unless:
It becomes compacted (the change is trivial for sands).
It is structurally deformed (normally such deformation is obvious).
It is bioturbated (reworking by organisms that may or may not leave a
visible structure).
a) How grain orientation is measured.
i) Gravel-size material.
Measured in terms of the a-axis (dL), the b-axis (dI) and the plane of
maximum projection (the a-b plane).
Measure the strike and dip of the a-b plane.
Strike will be the trend of the a-axis or the b-axis.
Dip is measured from the plane of
bedding (not the horizontal plane if
the beds are tilted).
In tilted rocks measure the dip
with respect to the horizontal and
correct for regional dip of bedding.
The particle will dip along the a- or
b-axes.
The dip of a particle is termed imbrication.
The direction is the imbrication direction and the dip angle is the angle
of imbrication.
On average, the a-axis is either parallel to or perpendicular to the
direction of imbrication.
Particles that are deposited
from water normally dip into
the current; imbrication
direction is at 180 to the flow
of the depositing current.
ii) Sand-size material
Can be determined with a microscope from thin sections cut from
oriented specimens.
Before removing the specimen from the outcrop it should be marked to
show:
1. The direction to magnetic north.
ii) Sand-size material
Can be determined with a microscope from thin sections cut for oriented
specimens.
Before removing the specimen from the outcrop it should be marked to
show:
1. The direction to magnetic north.
2. The top direction in outcrop.
ii) Sand-size material
Can be determined with a microscope from thin sections cut for oriented
specimens.
Before removing the specimen from the outcrop it should be marked to
show:
1. The direction to magnetic north.
2. The top direction in outcrop.
3. The attitude of associated bedding.
In the lab, three thin sections must be cut:
1. Parallel to bedding. Determine
the orientation of the apparent long
axes; this indicates the orientation
of the remaining two thin sections.
2. Parallel to the average long axis
orientation but perpendicular to
bedding.
3. At right angles to the average
long axis orientation and
perpendicular to bedding.
The thin sections allow the identification of the average a-axis orientation,
whether the a- or b-axes are imbricate and the direction of imbrication.
b) Types of Grain Frabric
i) Isotropic fabric:
No preferred alignment of
the particles.
If the particles are highly spherical
no preferred fabric will be
discernable.
Particles are non-spherical but
have no preferred fabric.
ii) Anisotropic fabric:
A-axes transverse to flow with
b-axes imbricate.
a(t)b(i)
A-axes parallel to flow with aaxes imbricate.
a(p)a(i)
Complex fabrics also develop with a mix of a(t)b(i) and a(p)a(i) that may
appear isotropic.
A problem with measuring grain orientation in thin section:
Like apparent grain size, the
orientation that is seen in thin
section varies with where the thin
section cuts through the grain.
Care must be taken to ensure that the
thin section is in the plane of bedding
and not at a high angle to it.
a) Displaying directional data.
Directional data come in a variety of forms including the examples listed
below:
Particle a-axis orientation;
Imbrication direction;
Cross-bed dip direction;
Symmetrical ripple crest orientation;
Orientation of fossils;
Sedimentologists normally display such data on circular histograms
called “rose diagrams”.
The histogram, below, shows the
distribution of directional data
relatively clearly.
A single mode (at 120 to150° )
with an approximately normal
distribution.
The data shown below appear to be
bimodal (one from 0 to 30° and one
from 330 to 360°).
The distribution appears to be very
different than normal.
Actually, the distributions are very similar and effectively normal but this
cannot be recognized on such histograms because 0 and 360 are equal
but shown to be at extreme ends of the scale.
Sedimentologists normally display directional data on a rose diagram:
A circular histogram with directional data grouped into directional
classes.
Florence Nightingale’s “Polar-Area Diagrams” or “coxcombs”
Mortality figures during the
Crimean War (1854 - 56)
30° class intervals
30° class intervals
30° class intervals
30° class intervals
30° class intervals
30° class intervals
Rose diagrams better display the distribution of directional data than
regular histograms because they give a sense of the spatial significance
of the data.
There are two common ways of showing the frequency (number of
observations) scale on a rose diagram:
Length Proportional Scale
Length of scale is proportional
to the number of observations
Area Proportional Scale
Length of scale is proportional to
the square root of the number of
observations; segment area is
proportional to the number of
observations.
Length Proportional Scale
Area Proportional Scale
1 1
=1
Rose segments:
Length Proportional Scale
Area Proportional Scale
2  1 .4 1
=2
Rose segments:
Length Proportional Scale
Area Proportional Scale
3  1.73
=3
Rose segments:
Length Proportional Scale
Area Proportional Scale
4  2
=4
Rose segments:
Length Proportional Scale
Area Proportional Scale
5  2.24
=5
Rose segments:
Length Proportional Scale
Number of
Observations
(n)
Area Proportional Scale
Segment Area
Segment Area for n=1
Number of
Observations
(n)
1
1
1
1
2
22= 4
2
2
3
32 = 9
3
3
4
42 = 16
4
4
5
52 = 25
5
5
In a graphical representation of the data the eye sees the area of the
segments.
With a length proportional scale the sense that is given is that an increase
in number of observations from 1 to 5 is 25 times rather than 5 times.
The area proportional scale shows an increase in area that is truly
proportional to the increase in the number of observations.
Length proportional scales overly emphasizes class intervals with large
numbers of observations.
Types of Rose Diagrams
Unimodal: with one prominent mode (predominant direction).
Bimodal: with two modes.
Bipolar: with 2 modes at 180° to each other.
Polymodal: with three or more modes.
Always make sure that you know what kind of data is being presented in
a given rose diagram.
Some data are unidirectional (point only in one direction; e.g., the dip
direction of a planar surface such as cross-bedding).
On a rose diagram for such data each observation will have one unique
direction.
Some data are bidirectional (a trend with two directions at 180° to each
other; e.g., the alignment of a particle long axis.).
Quite often bidirectional data are
plotted to show both directions
associated with the trend.
The rose diagrams will plot as what
appears to be perfectly symmetrical
bipolar distributions whereas the
data are actually unimodal.
Rose diagrams for common anisotropic fabrics are shown below
(note that all of the roses are shown as bidirectional data and are not
really bipolar).
b) Statistical Treatment of Directional Data
Directional data cannot be treated with scalar arithmetic calculations for
statistical measures because directional data are circular (vary from 0 to
360°) and not infinitely continuous.
E.g., what is the average of the
three directional measurements?
Treated arithmetically:
m ean 
346°  24°  67°
3

437°
 146°
3
Clearly wrong!
Directional data must be treated as vectors.
Every vector has two parts: direction and magnitude.
Think of a vector as an arrow pointing in some direction (q the lower
case Greek letter theta) and the arrow has a length (R) which is its
magnitude (the longer the arrow the greater the magnitude).
Every directional measurement is
a unit vector; a vector with a
magnitude equal to 1.
The average direction can be
determined by lining the unit
vectors up end to end and joining
The three directional measurements the beginning and the end.
are represented as:
The “average” vector is termed the
resultant vector (q ).
It points in the average direction of
the data.
It has a specific direction and a
magnitude:
In this case:
q  2 5 .5 
R  2 .5 2
The magnitude of the resultant vector depends on the amount of variation
in the directions of the directional observations.
A data set with wide dispersion of A data set with little dispersion of
directions.
directions.
Relatively small magnitude of the resultant.
Relatively large magnitude of the resultant.
Statistics for directional data can be calculated for
both grouped and ungrouped data.
Ungrouped data
Grouped data
The following outlines the steps to calculate the direction and magnitude
of the resultant vector:
Step 1. Calculate the direction of the resultant vector.
Ungrouped Data
Grouped Data
(raw data, values are individual measurements)
(number of observations per directional class)
N
w
n
i
NC
sin q i
w
i 1
n
i
sin q i
i 1
NC
N
v
n
i
v
cos q i
n
i
co s q i
i 1
i 1
Where: N is the total number of
observations; ni is the magnitude of
the ith vector (=1 for each
observaton); qi is the ith
observation.
1 w
q  tan
v
Where: NC is the number of classes;
ni is the number of observations the
ith class; qi is the direction of the
midpoint of the ith class.
-90 < q <+90
q  tan
1
w
v
-90 < q <+90
Apply the Case Rule to determine
the true value of q.
if w>0 AND v>0
q remains unchanged
if w>0 AND v<0 OR w<0 AND v<0
add 180° to the calculated value of q
if w<0 AND v>0
add 360° to the calculated value of q
Step 2. Calculate the magnitude (R) of the resultant vector.
R 
v w
2
2
Remember that R is some proportion of the sum
of the magnitudes of all of unit vectors in the
data set. Its value depends on the total number of
observations in the data set and the amount of
dispersion.
A useful measure of the dispersion is to express the magnitude of the
resultant vector as a percentage of the sum of the magnitudes of all of
the unit vectors in the data set (L), where:
R
L
 100
N
L  100%
There is little dispersion of the data (all
measurements point towards the same direction).
L  0%
There is a great deal of dispersion of the data
(measurements point in all direction).
L = 26%
L = 92%
Step 3. Calculate the probability that the directional data are
uniformly distributed (p).
p e

2
 1 L  N  0.0001)

e is the natural logarithm = 2.71828,
N is the total number of measurements in the
data set.
p = 1; an equal number of
observations in each directional
class.
p  0; all observations fall into
the same directional class.
Raw data:
Grouped data:
184
187
191
196
198
201
204
205
207
208
210
212
214
216
222
224
Class interval
Midpoint
Frequency
180-189°
184.5°
2
190-199°
194.5°
3
200-209°
204.5°
6
210-219°
214.5°
4
220-229°
224.5°
2
Note: 10° classes.
Total (N):
17
205
Raw data:
184
187
191
196
198
201
204
205
207
208
210
212
214
216
222
224
Treatment of ungrouped data:
N
w
i 1
sin 184°
n i sin q i  sin 187°
sin 191°
sin 196°
sin 198°
sin 201°
sin 204°
sin 205°
sin 205°
sin 207°
sin 208°
sin 210°
sin 212°
sin 214°
sin 216°
sin 222°
sin 224°
Total:
-7.04
N
v
i 1
cos 184°
n i cos q i  cos 187°
cos 191°
cos 196°
cos 198°
cos 201°
cos 204°
cos 205°
cos 205°
cos 207°
cos 208°
cos 210°
cos 212°
cos 214°
cos 216°
cos 222°
cos 224°
Total:
-15.13
205
N
N
w   n i sin q i   7.04
v   n i cos q i   15.13
i 1
q  tan
1
i 1
w
q  tan
v
1
 7.04
 15.13
 24.95°
 q  2 4 .9 5 °+ 1 8 0 °= 2 0 4 .9 5 °
Apply the Case Rule:
if w>0 AND v>0
q remains unchanged
if w>0 AND v<0 OR w<0 AND v<0
add 180° to the calculated value of q
if w<0 AND v>0
add 360° to the calculated value of q
N
N
w   n i sin q i   7.04
v   n i cos q i   15.13
i 1
R 
L
v w
2
R
i 1
 100
L
N
p e
  15.13 
R 
2
16.69
2
   7.04   16.69
2
 100  98.17%
17

2
 1 L  N  0.0001)

p e

2
 1 98.17 17  0.0001)
q = 204.95°
R = 16.69
L = 98.17%
p = 7.67  10-8

 7 .6 7  1 0
8
Grouped data:
Class interval
Midpoint
Frequency
180-189°
184.5°
2
190-199°
194.5°
3
200-209°
204.5°
6
210-219°
214.5°
4
220-229°
224.5°
2
Note: 10° classes.
Total (N):
17
Treatment of grouped data:
NC
w
n
i 1
NC
v
n
i 1
i
i
sin q i  2 sin 184.5°+ 3sin194.5°+ 6sin204.5°+ 4sin 214.5°+ 2sin224.5°
  7 .0 6
cos q i  2 cos 184.5°+ 3 cos 194.5°+ 6 cos 204.5°+ 4 cos 214.5°+ 2 cos 224.5°
  1 5 .0 8
NC
NC
w   n i sin q i   7.06
v   n i cos q i   15.08
i 1
q  tan
1
i 1
w
q  tan
v
1
 7.06
 15.08
 25.09°
 q  2 5 .0 9 °+ 1 8 0 °= 2 0 5 .0 9 °
Apply the Case Rule:
if w>0 AND v>0
q remains unchanged
if w>0 AND v<0 OR w<0 AND v<0
add 180° to the calculated value of q
if w<0 AND v>0
add 360° to the calculated value of q
NC
NC
w   n i sin q i   7.06
v   n i cos q i   15.08
i 1
R 
L
v w
2
R
i 1
  15.08 
R 
2
 100
L
16.65
N
p e
2
   7.06   16.65
2
 100  97.95%
17

2
 1 L  N  0.0001)

p e

2
 1 97.95 17  0.0001)
q = 205.09°
R = 16.65
L = 97.95%
p = 8.25  10-8

 8.25  10
8
Ungrouped data results:
q = 204.95°
R = 16.69
L = 98.17%
p = 7.67  10-8
Grouped data results:
q = 205.09°
R = 16.65
L = 97.95%
p = 8.25  10-8
Slight error in the Grouped Data method because the actual observations
are not used.
c) The Significance of Particle Orientation.
i) Particle imbrication indicates the
paleoflow direction: the current flowed at
180 to the imbrication direction.
ii) Some data suggest that with
increasing flow strength the angle of
imbrication increases.
These results are not verified in other
experiments.
iii) a(t)b(i) fabric develops when particles roll
over the bed prior to deposition.
iv) a(p)a(i) fabric develops when particles are
transported in suspension and/or rapidly
deposited from high concentration.
Requires a more powerful current for a given
grain size than a(t)b(i) fabric.
v) Complex distributions can be interpreted in
terms of multiple transport mechanisms.
The rose to the right can be interpreted as a
mixture of grains that rolled on the bed (modes
perpendicular to inferred flow direction) and
grains that were deposited from suspension
(modes parallel to the inferred flow direction).
vi) Variation in imbrication direction.
Thin section of sandstone
Average imbrication direction through a
sandstone was measured in contiguous
rectangular areas (2 mm x 0.1 mm)
River sands display imbrication that varies in
angle but not in direction; always dipping into the
depositing current.
Sands that were deposited in a shallow marine environment display
imbrication directions that vary cyclically back and forth indicating
reversing currents.
This reflects the prevalence of oscillating currents produced by waves in
the shallow marine environment.
```