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Transcript of SPE-28036-MS_Relationship Between Frictional Strength and Fault Gouge Generation From Direct Shear...
8/20/2019 SPE-28036-MS_Relationship Between Frictional Strength and Fault Gouge Generation From Direct Shear Testing Un…
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SPE 28036
Relationship between frictional strength
and
fault
gouge
generation
from
direct
shear
testing under constant normal displacement control
La relation entre
la
force de friction et la generation de mylonite microgrenue
de
faille a 'aide
d'un appareil acontrainte dirigee avec un controle constant du deplacement
normal
Eine Beziehung zwischen der GleitfHichenreibungsstiirke und Trtimmerpartikelerzeugung
unter Verwendung einer Abschermaschine unter normaler gleichbleibender
Verschiebungskontrolle
B. R.Crawford, SPE,ISRM
B.G. D. Smart, SPE, ISRM Department
of
Petroleum Engineering, Heriot-Watt University, Edinburgh, UK
B.T.Ngwenya, Department of Geology Geophysics, Edinburgh University,
UK
Copyright 1994. Society
of
Petroleum Engineers
This paper was prepared for presentation at the 1994 Eurock SPE/ISRM Rock Mechanics n
Petroleum Engineering Conference held n Delft, The Netherlands, 29-31 August 1994.
Abstract: The production of
rictional
wear
debris
fault
gouge)
under optimum non-dilatant conditions
conducive to maximum dynamic fragmentation
has
been undertaken utilising a novel direct
shear
machine.
A wide variety of
sedimentary
lithologies
have been
tested under
constant
normal
displacement
control and
a
mechanistic model
for
the
evolution
of
frictional
resistance developed.
X-ray
diffraction
quantitative
mineral identification
and
laser diffraction particle sizing technologies have
been
employed to characterise
the induced debris distributions. A relationship is presented
between
the frictional strength
o
the sliding
surfaces and the specific surface area
of
he induced cataclastic debris.
Resume:
La
production de debris
defriction
mylonite microgrenue defaille) a ere effectuee a
l aide
d un
nouvel appareil a contrainte
dirigee
sous des
conditions
optimales non-dilatantes conduisant a une
fragmentation
dynamique maximale. Une large variete de lithologies
sedimentaires
a ete
testee
avec
un
controle
constant
du
deplacement
normal
et un modele mecanique de
I evolution de la
resistance a la
friction a ere developpe. Les techniques
d
identification quantitative de mineraux par diffraction de
rayons
X et
d
evaluation
de la taille des
particules
par diffraction
laser
ont
ete
employees pour caracteriser
les
distributions
des
debris induits. Une relation entre la force de friction
des
surfaces coulissantes et la surface
specifique
des
debris cataclastiques est presentee.
Zusammenfassung:
Unter
Verwendung einer neuartigen Abschermaschine ist bei optimalen,
ausdehnungsfreien Bedingungen
die eine
maxima
e
dynamische Zerkleinerung
erlauben
Abnutzungsabrieb
HohlmeifJel) erzeugt worden. Eine Vielzahl unterschiedlicher Sedimentgesteine wurde unter
normaler
gleichbleibender
Verschiebungskontrolle getestet und ein
mechanisches
Modell flir die Entwicklung des
Reibungswiderstandes erstellt.
Fur
die quantitative Mineralbestimmung wurde das Rontgendiffraktometer,
fur die Ermittlung der Partikelgroj3en das Laserdiffraktometer verwendet,
mit
der die Verteilung des
Abnutzungsabriebes
charakterisiert
wurde.
Eine Beziehung zwischen
der
Gleitflachenreibungsstarke und
der spezifischen Oberflache der erzeugten kataklastischen Trummerpartikel wird dargestellt.
Introduction
Current visualisation of fractured
hydrocarbon
reservoirs with regard
to
flow
simulation has significantly advanced from the
sugar-cube ideal
with,
in
particular, the
effects
of
observed power-law
size populations
of faults
and
the influence
of small-scale sealing
faults
below the
limits
of
seismic
resolution, now being
quantitatively
incorporated into
reservoir
simulation
model gridblocks
(Heath
et al )1. Also, stress
sensitivity
has
been
introduced into simulation
studies.
During reservoir
development, the stress
field
can
be severely modified
by
transients
in pore
pressure and temperature resulting from production
or
injection.
Heffer
et al simulated the
effects
of
such stress changes on existing fracture
deformation,
and
predicted new fracture genesis.
Bawden
et al
3
were
amongst the
first
to show
that,
contrary to the assumption of perfect rigidity,
fracture deformability could
be
an important
influence on fluid flow.
They
investigated the
effects of normal and shear deformation on
fracture
conductivity using
the
finite
element
technique,
assuming a
straightforward
coupling between
dilatancy
and
conductivity.
The
model
was found to
be very sensitive to input data, especially for
dilatant
shear problems,
depending on a complex
interaction
between normal
load,
maximum
fracture
closure,
normal
stiffness
and the geological (stress)
history. However,
with normal movement
becoming progressively
suppressed under
increasing normal stress and
hence
with
increasing
depth,
dilatant
shear
within many
hydrocarbon
reservoirs may
well
be a second
order parameter,
and
indeed may be wholly repressed
under some
in
situ conditions. Despite this, the
vast
majority of
experimental discontinuity shear testing
is
2
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conducted within portable shear
box
type
equipment modified from the soil mechanics
field,
under low magnitude, constant normal
stress
conditions (zero normal stiffness mode) more akin
to
surface structures where the possibility
of
significant
dilation exists.
In keeping with the maxim that, the
powerful numerical simulators currently available
are only empty
shells
if
they
cannot be
filled
with
reliable
data, a series of direct shear
tests
have
been
conducted
in order
to
assess the
frictional
strength characteristics
of
fractures
subjected to
shear displacement under constant normal
displacement (zero
dilation
equivalent
to infinite
normal
stiffness)
conditions. Frictional strength
parameters derived
from the
direct
shear
programme
have been quantified for a
wide
variety
of sedimentary lithologies, and are envisaged as
representing
input data for coupled fluid flow
and
deformability reservoir simulation studies. Shearing
under high normal stiffness represents the optimum
conditions
for asperity fracture and comminution,
and
as
such might well offer
a
more realistic
analogy for
sealing
faults, in
which the
principal
mechanism
of porosity collapse has
been
cataclasis
and granulation (cataclastic slip bands
or
granulation seams). Other areas
of
applicability lie
within the
near-wellbore environment related to
stability, sand production and stress-induced skin
effects.
Theory
The n
situ frictional
response of a rock
discontinuity to shear loading is dependent, not
only on material parameters such as surface
roughness
and asperity hardness/toughness,
but
also
on the
boundary
conditions operative across the
fracture walls.
These boundary
conditions can
be
variable.
Saeb Amadei
4
represented the range of
discontinuity-normal loading conditions
by
modelling the deformability
of
the surrounding
rockmass
as a
spring
with
normal
stiffness, k
n
=
dan/dOn, where dan and d n
are the changes in
discontinuity normal stress and displacement
respectively.
The
stiffness,
k
n
can
thus
be
considered to vary
between zero
for a
discontinuity
under
constant
normal
stress (as in slope stability
analyses)
or in
a
highly compliant rockmass,
and
infinity for
a
discontinuity in
a
highly
stiff
rockmass
where
no
change in
discontinuity-normal
deformation
is possible (as for tightly constrained
fault
blocks). Whilst
constant
normal stiffness
conditions may result if the change in normal
stress
and
normal displacement
remain
proportional,
stiffness
may
also vary with loading
history.
Although
finite, constant
or
variable
stiffness,
discontinuity-normal boundary conditions
are more likely to exist
n s tu
than either the zero
or infinite k
n
states, constant
normal
displacement
shearing is of particular interest, representing
optimum conditions for dynamic fragmentation and
wear debris (fault gouge) production by the process
of
cataclasis.
However, although Goodman
5
devised a method of predicting the
shear
response
of discontinuities under
constant
normal
displacement, based
on
a knowledge of
their
normal
loading behaviour and shear response
under
constant normal stress, as far as the authors
are
aware
no
experimental
results have
been
published
on direct shear testing under constant normal
displacement control. Johnston et l 6 developed a
constant normal stiffness direct
shear machine,
and
demonstrated an
increase in
measured shear stress
on failure with increasing normal stiffness, from
which it
can
be
assumed that the shear resistance to
slip under the constant normal displacement
criterion
will
be
very
different
from, and
significantly
greater
than that appropriate to the
fixed
normal
stress
mode.
In
this
study,
the mechanism of frictional
sliding under minimum dilation,
maximum
fragmentation
conditions associated with extreme
normal stiffness boundaries, is quantified
experimentally for a wide range
of
sedimentary
rock types,
using a novel
new direct shear rig,
capable of imposing a
constant
normal
displacement state. Also, X-ray and
laser
diffraction
technologies have been applied
to
thoroughly
characterise
the
large quantities of resultant
comminution products, the latter facility
representing a new capability not yet fully
utilised
by
the
petroleum industry. Through quantification
of the shear debris
sub-micron
sized fractions,
achieved via
laser
particle
sizing,
a
relationship is
demonstrated between sliding surface frictional
strength and a
measure of
the
induced
particle size
distribution.
The
value
of
such small particle
statistical
analyses
is
corroborated by
a study of
induced shear fracturing under triaxially
compressive stress states, in
which
the permeability
of
faulted core plugs
was
found to correlate
with
the
fractal dimension of the produced gouge, measured
by the laser
particle sizer
(Crawford et l
7 .
Equipment
A scaled
schematic
diagram of the
servo
hydraulic direct
shear
rig is shown in Fig. 1. The rig
was fabricated by RDP-Howden
Ltd.,
U.K., to a
customer-specified
design.
The
frame consists of
two
steel
side plates with a fixed base,
and
an
adjustable top
beam on
which a
500kN
hydraulic
linear actuator with a vertical stroke of
200mm
is
mounted, to
provide vertical normal reaction. The
vertical
actuator piston
rod is
attached
to an upper
shear box. Specimen sliding
displacement,
up
to
a
maximum of
100mm,
is achieved by means of a
unique
design
of
shear
table
which
utilises
a
linear
hydrostatic bearing to provide
effectively
frictionless motion,
even when acted on by the
500kN
normal compressive load. The
shear table
is
attached to a
lower shear
box.
Specimen
shearing
is
achieved
by relative horizontal
displacement
of the
lower shear box
fixed to
the
base table,
with respect
to a stationary upper shear box, attached to the
122
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piston
head of the vertical
ram.
Sliding
displacement is generated by two horizontal,
opposing 500kN linear actuators with strokes of
l00mm,
each ram
acting only
in
push.
A constant pressure oil
supply
for the
vertical and horizontal
actuator
servo
systems
is
Floating Tahle Control:
T r a l 1 ~ t l l l c c r Imlicat()fS
provided
by
a
mobile
hydraulic power pack
with
a
maximum system
pressure
of 30MPa.
This power
pack
also supplies a lOOT
stiff
servo-hydraulic
compression rig and pressure intensifier
for
conventional
triaxial
compression
testing of core
plugs. The
shear
table
linear
hydrostatic bearing
is
Shear Rig Front Elevation:
~
~ .
I--
flormal 1I1S1 aC l lCIIl
0
.(1
I amPlificrl
(L VDT'S) vertical
hydraulic D ctuator
accumulator mounted
y
R
on
Vertical
Ram
crosshcad
PRESSURE
Control:
P-
INTENSIFIER SL2000
J
Lr _ _ _iJ
COMMAND
- ~
OM P:I'a';S ION
MODULE
:1
~
Hydraulic : ; T : b i ~ :
r ~ l ~ a ~ ~ )-1
for F o3lilu
(Ioadccll)
~ m { t ~ ~ H
fixing
pOSiti()n7
1
shear lox
for crosshead _I>
specimen holder
floating ~ a l l e on
I l
~
discontinuity
hydrostatic hearing
l ~ = ~
shear surr ce
19
9
~ h ~ a r
power
pack
aCluator9
-
shear load (pressure
shear u
.. , u ·
-
(LVDT'S)
Shear Rig Back Elevation:
\P)
c
r:
~
R
normal load 2 /
Ire
l ;u::rp
pressure .. ,
{
servo valve
-
0
0
0
SERVO-HYDRAULIC CONTROLLED, STIFF,
P
IRECT SHEAR l\IACIIINE
-
I
,
I
I I I I
I
I I
~ ~
I
(metres)
1
I Iydraulic Supply
for
Actuators:
~
C O M P ~ E S S I 0 1
(
PRESSURE
J
servo valve
INTENSIFIER
_ I ~ h
•
•
(
POWER
PACK
I
hydraulic oil
~ s s u r e P
-
ydraulic oil
return, R
~
l
-
#
Fig. 1 A
scaled
schematic of the direct shear rig,
123
=-
II
shear \
supply
I
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fed from
a
separate
mobile
hydraulic
power
pack
which
supplies
a
maximum
system pressure of up
to
20MPa,
and scavenges
the
spent
oil.
The bearing
action is
created by
feeding hydraulic oil
through
four orifices on
the
underside of
the shear table.
The 20MPa
system
pressure is dissipated
in
a
0.05mm gap between the rims of the orifices and a
lower
bearing
plate. The linear
actuator
providing
vertical thrust is suitable for operation under closed
loop control with displacement or pressure as the
controlling
parameter.
An
internal linear variable
differential transformer (L VDT) provides
measurement
of
piston travel for normal
displacement control, whilst
a
pressure
transducer
in the hydraulic feed line enables the actuator to
be
operated
under the alternative condition
of normal
load
control. The pair
of
single acting (push-only)
horizontal actuators
mounted either
side of the
hydrostatic shear table are controlled separately
from
the vertical ram. L
VDT'
s allow measurement
of horizontal displacement, whilst pressure
transducers in
the hydraulic feed-lines enable
measurement of shear loads. Hardcopy data from
tests performed under constant normal
displacement conditions
is
provided
in
the form
of
continuous line traces
of
normal
load,
shear load
and
normal displacement
against
shear
displacement, recorded
on a multi-channel,
moving-paper
chart-recorder.
To
characterise
size distribution of the
fragmented
frictional
wear products, a
Mastersizer laser
light
scattering-based, particle
size analyser, commercially available from
Malvern
Instruments
Ltd.,
was
employed.
As
far as is
known,
this
present study is the first to employ laser
diffraction technology to
assess
fault gouge. In
essence, the laser light scattering sizer comprises an
optical measurement unit that forms the basic
particle size sensor, and a computer
that
manages
the measurement and
performs results
analyses and
presentation.
The
system
utilises
composite laser
light diffraction patterns produced by the dispersed
particles to compute
size distributions.
The
Mastersizer
has
three standard user-defined
size
ranges:
600 - 1.2, 180 - 0.5 and
80
- O.lJ..lm. t
cannot
simultaneously
cover the
complete dynamic
range in one
measurement,
so the total span is
broken down into the above
size
ranges, each
selected by
fitting the appropriate
range
lens, of
focal
length 300mm,
100mm and 45mm
respectively. The
Mastersizer employs two forms
of
optical configuration to
achieve
its
wide
range.
Whilst conventional
Fourier
optics are used
for
the
two greater focal length
lenses,
the 45mm lens
utilises
a
reverse Fourier
optical
configuration to
minimise aberrations associated with large angle
scattering detection
(typically
>10
0
) from small
particles.
When
a particle scatters light (sourced
from
a low power
Helium-Neon laser) it produces
a
unique
light
intensity characteristic with angle
of
observation.
This light
is scattered
so that
the
measured energy on a detector has a peak at a
favoured scattering angle which is related to the
24
particle
diameter.
Large particles have peak
energies in small angles of scatter and
vice
versa, as
illustrated in Fig 2.
Over
the
size
range 2J lm and
upwards, the
scattering with angle is effectively
independent of the optical properties of the material
or supporting
medium,
resulting
primarily
from
diffraction of light around the particle, however at
the O.IJ..lm size range, interaction of
light
with the
particle is complex and strongly influenced by the
optical properties of the particle, necessitating full
Mie
theory
modelling
of
the
scattering (Mie
theory uses the solution of the electromagnetic
wave
equations
for a
spherical
particle
of
specific
optical
properties). Two
optical
constants
are
required
to
determine
the scattering
behaviour, the
differential refractive index
between the dispersant
and the particle, and the particle absorption, both of
which
are user
selectable.
Thus
it is possible
for
the computer to predict
the scattering
signal
that would be received from a
wide range of materials at a
given
size.
t
formulates
a table
that characterises
how a unit volume of
material, of a range of sizes throughout the working
range,
scatters light. Using
this
theoretical data the
computer deduces
the
volume size distribution
that
gives
rise to
the observed scattering
characteristics,
by
a
process of constrained least squares fitting of
the
theoretical scattering characteristics to
the observed
data. This
best
fit
result can
either
be obtained
with
no
assumed
form
of
size
distribution (model
independent) allowing
the
characterisation of multi
modal distributions
with high
resolution, or else the
analysis
can
be constrained to
a known form
of
distribution
such
as the
Rosin-Rammler
or
Log
normal in volume.
Results
A
dozen 118mm
=4.5 ) -diameter
specimens
sourced
from
blocks retrieved from
surface mining operations, were tested in direct
shear under
constant normal displacement mode.
A
comprehensive range
of
sedimentary lithologies
were tested including Permian red sandstone
(specimens #3, #5, and
#6) and Carboniferous
fissile mudstone (#8 and #9), carbonaceous
Large particles
scatter at
small angles
Small particles
scatter at
high
angles
Central
Detector
Radii
Fig Properties
of scattered
light for large and
small particles.
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Sliding Displacement
(mm)
Fig. 3 Shear
and
normal stress versus
sliding
displacement
profiles,
specimen S I (slip
dependent contact area).
5
c
Pre-Peak
+
Post-Peak
4
2
3
'
'
Ci
2
'
..c
n
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
o 2
3
4
5 6
7
Normal Stress (MPa)
Fig. 4 Frictional strength parameters under non
dilatant
maximum
fragmentation conditions,
specimen #8.
laminated
siltstone (#11,
#12 and #14) and
heterolithic sandstone/siltstone (S1,
S2, S3 and
S4).
Each
specimen was tested dry and
under
a
pre
shear initial
normal
load
of less
than
3MPa,
to a
maximum shear displacement of around 25mm,
with normal displacement (generally <O.1mm)
normal load and shear load variations being
recorded as synchronous traces
on
a chart recorder.
Details of
the
experimental methodology and
example load-displacement traces for the four
general rock
types
are given in Smart
Crawford
8
.
Shear and normal
load
curves were converted to
the
appropriate
stress
profiles by dividing by a
slip
dependent contact area, corrected to allow for
progressive decrease in
mated
shear surface area
with
increasing
shear displacement. This
more
realistic variable
contact area
analysis, as
opposed
to
the assumption
of a
constant contact
area,
was
observed to have a
first
order
influence
on
calculated stress profiles, as shown in Fig. 3 for
specimen S 1. In shear stress
versus
normal stress
space, all lithologies exhibited the same overall
attributes irrespective of rock
type,
namely a
non
linear increase in pre-peak
strength
data to a
maximum
friction
value (designated
Ii*max,
and
defined following Amonton's law as the peak shear
load attained divided by the corresponding value of
normal reaction) with post-peak data delineating a
robust straight-line relation with high correlation
coefficients,
defined
by a slope (designated the
coefficient
of
sliding friction, lis
and an intercept
(designated the inherent cohesion of the contact
surfaces,
So).
These relations
and
the
frictional
strength parameters Ii*max, lis
and
So
are shown in
Fig. 4, for specimen
#8.
Tabulated
values
for
comparison
of relative
magnitudes
between
the
different
rock types are
given in
able 1.
With
regard
to a
mechanistic explanation
for these
observed phenomena under constant
normal
displacement conditions,
maximum friction is
interpreted as
representing
peak shear strength of
able Correlation of frictional strength, composition,
and
fault gouge distribution parameters.
Direct Shear Testing
XRD
Analyses
Particle Size Analyses
Friction
Parameters:
%
Mineral
Content: Fractal
Dimension,
D = 3 - n:
I.D.
J.l*max
I
J.ls
I
So
Qtz
I spar I Clay I Mica
Sieving
I
=300mm
I
=45mm
Sw
(MPa)
(cmI\2/g)
#3 0.50
0.36
0.16 94.5
5.5 0 0 1.05
2.12
2.07
1891
#5 0.62
0.50
0.54
82.5
17.5 0 0
1.11
2.15 2.01
2008
#6
0.65
0.48
0.42
--- ---
---
---
0.80 2.20
2.13
2487
#8 0.69
0.43
0.83
15.1 38.8 46.1 0 1.80 2.31
2.05 1319
#9
0.82 0.43 0.38
0 0
100
0 1.83
2.26 2.03 1202
11 0.54
0.15
0.79
---
---
---
--- 1.88 1.46
1.63 874
#12 0.69
0.30
0.82
76.4 4.6
14.5
4.6
1.77 1.95 1.91
891
#14
0.64
0.15 0.98
65.4 8.7 22.1 3.8
1.76
1.88 1.67 454
Sl
0.88
0.21 1.93
44.9 5.7 38.7 5.7 2.05
2.00
1.95
1543
S2
0.62 0.35 0.95 44.5
4.1 45.5
5.9
2.26 2.05 1.85
1340
S3 0.49
0.26 0.76
24.4
0
70.7
4.9
2.14
2.13 1.84
1231
S4
0.80 0.39
2.13
45.1 0 45.5 9.4 1.95 2.08 1.88
1271
125
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the sliding
discontinuity, immediately
prior to
through-going
breaching
of the
interlocking
asperity system (usually associated with a
significant, and frequently audible
dynamic
stress
drop)
followed
by
frictional
sliding on
a newly
created, pervasive shear
plane.
Post-mortem
examination reveals
this
surface to
be
composed of
frictionally
comminuted wear debris (analogous to
natural
fault
gouge) frequently exhibiting steps and
slip-parallel striae (analogous to slickensides and
related features as observed on natural fault
surfaces).
Following
direct shear testing, the specimen
halves
were
separated and a small sample of debris
(covering approximately 5% of the initial pre-shear
contact
area)
removed
from corresponding patches
on the upper (TOP) and lower (BOT) sliding
surfaces.
These
samples were
then
sieved
independently using B.S. Sieve No.'s 30 (500f..lm),
36 (422f..lm), 52 (295f..lm), 60 (251f..lm), 100
(152f..lm), 120 (l24f..lm),
150
(l04f..lm) and 170
(89f..lm). The
remaining
debris from both surfaces
was then added together and the total sample sieved
into >500f..lm and <500f..lm fractions, the former
being used
for
X-ray diffraction
(XRD) analysis to
characterise percentage mineral
content
(Table
1)
the latter
for
laser diffraction particle sizing
utilising both
the
300mm
and
45mm focal length
lenses
(resulting
in particle distributions by volume
from
600 - 1.2f..lm
and
80 - O.If..lm respectively).
Such fragmentation
data
has been
analysed
in
terms
of the Rosin-Rammler
9
distribution function, an
empirical, two-parameter, Weibull-equivalent
relation originally developed for broken coal, in
which for the residual
weight
distribution:
Eqn.l
In Eqn. 1, R is the cumulative percent
oversize,
x
the particle size
and
n a n d
b a r e
constants. Taking logs
to
the
base
10, this
represents a straight line relationship between log
log
100 R
and
log x. For theoretically
perfect
grading so that all fragments are the same size, n
00
for
steady size reduction n
=
1, and
for
more
rapid
diminution in size, n < 1, characteristic
of
a
dusty
material.
The constant n is
thus
a
measure
of the closeness of grading, it
fixes
the distribution
of particle
size, and
is therefore called the
distribution constant . The absolute size
constant , X, measures the actual
size
of
the
material, large values meaning
coarse
material and
small
values fine material.
Turcotte
10
,
related
the
power-law
mass distribution
given
in Eqn.l
to
the
definition
of
a fractal
number
distribution, and
showed
that for the fractal dimension, D:
D = 3 n
Eqn.2
Fig. 5 shows both the
sieve
and laser
particle sizing data plotted
in accordance
with the
linear
form of the Rosin-Rammler law, for
Laser sizing:
x
f=300mm
m
f=45mm
0
Sieving:
......
•
Bar
-
eo
0
TOP
..9
-1
6 '
0
-
-
-2
o
g
eo
..9
-3
EEl
.
IDD I
-
4
.1
10 100
1000
Particle
Size
(microns)
Fig. 5 Power-law relations for sieving and laser
particle
sizing volume distributions,
specimen 3.
specimen #3. For
both
the conventional and reverse
Fourier optics configurations, the resultant volume
distributions
show quite marked
sigmoidal
profiles,
with greater slopes at the upper
and lower
ends
indicative of more closely sized products. However
for specimen #3, the central
portions
of each plot
are
linear
and
broadly
parallel,
indicative
of a
power-law distribution.
The best-fit
straight line to
such
portions of each
curve were
used to
define the
distribution constant, b, from which Eqn.2. was
used
to
generate an
inferred fractal dimension,
D,
as listed
in
Table 1.
For the
analyses
by
sieving,
straight
lines
were fitted to the entire size range
(500
- 90f..lm)
and the upper- and
lower-sliding
surface results
averaged to arrive at
aD-value.
The
sigmoidal shape to
some
distributions
may
well reflect
complex
fragmentation
occurring
within the
mUlti-component
specimens, different
mineral species possessing different specific
surface
energies and crystal structures.
Whilst
scale-invariance is
inherent in
the
definition of
a
fractal or
fractional
dimension, natural
as
opposed
to
mathematical fractals do possess
finite upper and
lower
fractal limits above
and
below which
this scale
invariance
breaks down.
Samples
with
clay contents
approximately
~ 5 0
(specimens #8, #9, S2, S3 and S4) show strong
convergence
of both
the
6OO-1.2f..lm and 80-0.1f..lm
size range distributions at around the If lm scale,
suggestive of the attainment of a critical lower
limit to
clay
fragmentation.
For
all samples, good
agreement (relative parallelism) is shown between
the distributions
by
sieving
and
the
300mm
focal
length laser sizing data over
the
equivalent size
range.
Discussion
126
Grady
Kippll
proposed that growing
evidence from a wide variety of different
fragmentation
methods, whilst far from being
complete or systematic, did tend
to
suggest that
the
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broad range in size of the distribution constant, n,
might be related,
at least in
part,
to the
type
or
method of fragmentation. Several different
fragmentation processes are listed below in order of
decreasing n:
fragmenting munitions (bomb casings)
direct impact fragmentation (spalling)
Hopkinson bar
testing
(torsional)
Interplanetary debris
(asteroid
breakup)
Ball
milling
(multiple comminution)
6 4
3 2
1.8 - 1.2
::::1
1 -
0.6
From the above list, Grady Kipp
suggested that
single tensile
fragmentation
leads to
fragment
size distributions
with large values
of
n,
whereas
in contrast, fragmentation
with
significant
shearing and continued comminution
leads
to lower
values. In
Fig.
6, average
distribution
constants, n,
from the relatively large-size
fraction
sieving
analyses
are plotted against percentage
mineral
content,
determined
by
XRD of the
multilithologic
direct shear specimens. From this,
it is
evident that
the distribution constant
increases
fairly
systematically from
n :::: 1
to
n
=
2
with increasing
quartz content, and vice versa with increasing clay
content. As the
method
of
fragmentation has been
the same for all rock types, that
of direct
shear
testing under constant
normal displacement
control,
this
observed variation in distribution constant
therefore must
reflect
material
property
differences.
A
tentative
explanation can be
found by
considering
the
micromechanics
of
the
deformation. For
the
high
quartz content
sandstones
with grain-supported ball-bearing like
structures,
macroscopic shearing
will on
the
microscale
be
transmitted as
stress-concentrations
at the point-to
point quartz grain
contacts,
resulting in
predominantly tensile splitting exactly
analogous
to
axial
microcracking in
triaxial
compression
specimens. By contrast,
due
to the foliated nature
of
the clay-rich
lithologies,
macroscopic
shearing
may
activate this fabric
resulting in
foliation shearing as
I=l
~
n
I=l
0
U
§
.g
D
.5
on
is
U
U
;>
<:
2.5
2.0
I
. Quartz
I
• Clay
1.5
•
1.0
b
0.5
0.0 +-...-.--.--.- 'T-r-....-...-,........., .......-,..--.-....,....,-..--,.........-....-i
o
25
50
75
100
Percentage
Mineral Content
Fig.
6 Dependence
of
fault
gouge distribution
constant on source
lithology mineral
content.
observed in natural argillaceous fault gouge. Thus
the
mechanisms of strain accomodation on the
microscale are determined by initial rock
microstructure,
with the
relative
amounts of
resultant
shearing fragmentation to
tensile cracking
being reflected in the debris distrbutions.
As the
surface area of a bed of
powder is
inversely
related to
the
particle size, the
lowest
size-range distribution (laser particle
sizer
with
45mm focal length lens) yields the greatest estimate
of specific surface area, calculated according to
Herdan
12
, by:
Eqn.3
Sw is the specific surface area per unit
weight, p is
the
density of
the
mineral phase, wdh
is the harmonic mean of
the
weight distribution,
and
the 6
corresponds to the shape factor
of
the
particles
(the laser sizer assumes they
are
all
spheres). Tabulated values of
Sw are
given in
able
1, from which it is apparent
that the
sandstone lithologies generate the largest surface
areas of
fault
gouge.
An
interesting direct
correlation
is
shown between
the
specific surface
area,
Sw,
and the coefffcient
of
sliding
friction,
J.L
s
(Fig.
7) the
relatively
slippy carbonaceous
siltstones possessing low frictional coefficients and
generating low shear debris surface areas,
the
red
sandstones vice versa.
The
Sw
versus J.L
s
relationship
can
perhaps
be rationalised through consideration of
the
following:
For
each
direct shear test, an
empirical
Coulomb-type linear
friction
law has been
demonstrated for sliding on newly created,
slickensided
and
shear
debris
covered surfaces:
Eqn.4
However
implicit
in
this formulation
is
the
assumption of a p e r f e ~ t l y - m a t e d
contact area
127
0.75 . - - - - - - - - - - - - - - - - - . . . . .
S3
I SI
0.00
- t - r - . - . ~ . . . . ........................ .. .... .... .... ..... .... .............. . . . . . . ~
....
..... ..........j
o
500 1000
1500
2000
2500
3000
Specific
Surface
Area
cmI\2/g)
Fig.
7
Relationship
between
frictional
strength
and
fault gouge specific
surface
area.
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(although
corrected
for
the
effect of shear
displacement) when converting the measured shear
forces and normal reactions to stresses. Thus an
apparent contact area.
Aapparent
is assumed which
will
be
greater
than the
actual contact
area. Area .
occurring at the tips of touching asperities
associated with
rough
surfaces
in contact. Thus in
Eqn.
4.
the
shear
and
normal stresses can be
annotated
with the subscript apparent'·.
The
actual
stresses.
'treal
and
O'reah are
given
by
multiplying
the
calculated stresses. 'tapparent
and
O'apparent by
the
ratio (Aapparent / Areal). On
rearrangmg. Eqn.
4
becomes:
_ S (Aapparent)
'treal
0
A
l iS = real E 5
-
• qn.
O real
In Eqn.
5
above.
J 1
s
oc -IIA
rea
l.
so
that
as
the contact
area increases.
the frictional coefficient
also increases.
For
post-maximum friction
conditions
under
infinite
normal
stiffness. residual
sliding
is occurring wholly upon
a
bed
of
powdered
rock fragments. so that the
contact
area
is
best
described. not by fault wall
roughness
and asperity
contact.
but rather by the
specific
surface area of
the debris bed.
Sw =Areal. For smaller particle
size
and
therefore
greater
surface area. the
real area of
contact tends
more and more
towards the assumed.
perfectly-mated
contact area. the
ratio
Aapparent:Areal
decreases and
from
Eqn. 5. the
frictIonal
coefficient increases. thus
explaining
the
relationship in
Fig.
7.
Conclusions
A
novel
servo-hydraulic direct shear
machine
operated under constant normal
displacement (maximum normal
stiffness)
control
has
been used
to test lithologies ranging
from
sandstones
to
mudrocks. Once interlocking
asperities
have
been sheared through. residual
sliding
takes place
on a new
pervasive
shear plane
of
comminuted rock debris.
with
frictional strength
conforming
to
a linear Coulomb-type>'
criterion.
Sliding coefficients are less than (down
to
0.15)
and cohesions
greater
than
(>2MPa)
those
usually
measured
under dilatant. constant normal stress
conditions.
Such data
is
considered
relevant to
coupled
fluid flow
and deformability
reservoir
simulation in
which
sealing faults are expressly
incorporated. t is also directly pertinent to the
near-wellbore environment. Induced shear debris
volume
distributions
have been
quantified
down to
0.1 J.1m using new laser diffraction technology. such
analyses proving valuable in understanding both
macroscopic strength
and
the micromechanics of
deformation.
Future experimentation should
focus
on
characterisation
of both constant normal
stiffness and dilatant shear frictional response.
ideally
in combination
with
development of a fluid
flow
capability to enable simultaneous
permeability
quantification
during
shearing.
128
Nomenclature
A:
D
wdh:
k
n
:
n:
R:
So:
Sw:
x:
x:
on:
J.1*max:
J 1s:
O n:
't:
p:
Subscripts:
direct
shear
specimen
contact area
fractal dimension
harmonic mean of weight distribution
normal
stiffness
Rosin-Rarnmler distribution constant
residual
weight
%
cohesion of sliding surfaces
specific surface area
per
unit weight
particle size
Rosin-Rarnmler absolute size
constant
normal displacement
maximum
friction
coefficient
coefficient
of
sliding
friction
normal stress
shear stress
density of single mineral phase
apparent: assumed perfectly-mated contact area
real:
true rough contact area
References
1
Heath.
A.E
•
Walsh.
J.J..
Watterson.
J.:
Estimation of the
effects of
sub-seismic
sealing faults on effective permeabilities in
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reservoirs.
In:
Aasen
et
al
(Eds.)
North
Sea Oil and Gas Reservoirs - III
(1994)
NTH.
Kluwer
Academic Publishers.
Ch.
9. pp.173-183.
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K.J • Last. N.C.. Koutsabeloulis.
N.C
•
Chan.
H.C.M
•
Gutierrez.
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A.:
The influence
of natural fractures, faults and
earth
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geomechanical
analysis
by numerical
modelling;' In:
Aasen
et al
(Eds.)
North
Sea
Oil
and Gas Reservoirs - III (1994)
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Amadei. B.:
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York
(1989) 562pages.
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Johnston.
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•
Lam. T.S.K
• &
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pile design
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89.
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Crawford. B.R.. Main.I.G.
Smart,
B.G.D.:
Influence of the fractal structure of
fault
gouge on fluid
permeability
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during
deformation,
(1994) J
Struct. Geol.
(in prep.).
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B.G.D.
Crawford, B.R.: An
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debris
comminution as
a
mechanism
of
strain energy release for
frictional sliding
on dominant parting
planes,
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Brummer,
R. (Ed.)
Static
and Dynamic
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A.A.Balkema, Rotterdam, pp.389-400.
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Turcotte,
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Fractals
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29