Rock mechanics for engineering geology part 3

Rock
mechanics
Estimation of initial
Stresses

What Is Strain?
• Increase (or decrease) in
length resulting from a stress
acting parallel to the
longitudinal axis of the
specimen.
• strain is defined as extension
per unit length.
• Strain = extension / original
length

Stress-Strain Curve for Textile Fibre

Typical regions that can be
observed in a stress-strain
curve are:
• Elastic region,
• Yielding,
• Strain Hardening,
• Necking and Failure
Stress-Strain Curve for Textile Fibre

Elastic Behavior
• If the specimen returns
to its original length
when the load acting on
it is removed, it is said
to response elastically

Yielding
• A slight increase in stress
above the elastic limit will
result in permanent
deformation. This
behavior is
called yielding
• The stress that causes
yielding is called yield
stress sy.
• The deformation that
occurs is called plastic
deformation

Strain Hardening
• When yielding has ended,
a further load can be
applied to the specimen,
resulting in a cure that
rises continuously but
becomes flatter until it
reaches a maximum
stress referred to
as ultimate stress, su.
• The rise in the curve is
called Strain Hardening

Necking & Fracture
• After the ultimate
stress, the cross-
sectional area begins to
decrease in a localized
region of the specimen,
instead of over its entire
length. The load (and
stress) keeps dropping
until the specimen
reaches the fracture
point.

Initial stress
Stress state in rock mass without
artificial disturbances.
One of the basic data in designing
rock structure.
Competence factor = Uniaxial
compressive stress / initial
vertical stress is a good index for
stability of an opening in
rock mass
Initial stress

Stress state near opening is
disturbed
Disturbed zone is avoided
for measurement of initial
stress
Disturbed stress is also
measured in some cases
where investigating
deformation and failure
behavior around the
opening

Primitive Estimation
σ V = γ h
stress σV is
weight of
rock
Initial vertical
equal to the
overburden
(overburden pressure)
γ is unit volume weight
(ex. 27 kN /m3）
h is the depth (m)
Ex. Initial vertical stress
at 500 m deep is 13.5
MPa
σ V = γ h
σ h = k2σ V
σ H = k1σ V
Initial stress value

Initial horizontal stress σH at a shallow depth.
Horizontal stress ε is assumed to be zero.H
Eε H = σ H −ν (σ H
σ H = kσ V
+ σ V ) = 0
ν
k =
1−ν
E is Young's modulus.
ν is Poisson's ratio
k is the coefficient for horizontal stress
k is assumed to be 1 at great depth.
σ V = γ h
σ h = k2σ V
σ H = k1σ V

Measured initial stress values
Initial vertical
stress is roughly equal to the
primitive estimation
Initial horizontal
stress is different from the primitive
estimation

Average initial horizontal stress
σHavLower limit of (2.7 + 0.0081 h)
σHavUpper limit of (40.5 + 0.0135 h)
0
200
Average initial horizontal
stress = depth independent
value + depth proportional
value for 0.25 - 0.33 of
poisson's ratio
400
600
800
1000
0 10 20 30 40 50 60
Stress (MPa)
Depth(m)
σV

Depth independent value?
Movement of the tectonic plates
Spherical shell subsidence model
"Reprinted from http://pubs.usgs.gov/gip/earthq1/fig1.gif with permission from USGS".

Orientation of the maximum horizontal stress measured by
hydraulic fracture method (Goodman, 1980)
Average focal
mechanism of deep earthquakes in and around Japan projected on
the upper hemisphere. Arrows show tension and compression axes
(Kasahara, 1983)

Spherical shell subsidence model
The earth shrinks by
gravity force.
Compressive strain
appears by the spherical
geometry.

Need for initial stress measurement
Initial stress can roughly estimated by the primitive
method
Measurement is required for precise values
Initial stress can be affected by such geological phenomena as fold,
faults, intrusion of magma etc.
Method to measure initial stress

Method Description Feature
Stress relief method A borehole is drilled to desired depth. A
probe is installed in the hole. Stress
around he probe is relieved by usually
overcoring.
Three dimensional stress state can be
estimated by one overcoring in most
methods. It takes costs and time. There
are many results. The hole is drilled
usually from a roadway.
Stress compensation method Stress is relieved measuring displacement
or strain. Stress is applied until the
displacement or strain recovers to the
values before the stress relief. Necessary
stress is regarded as initial stress.
Measurement is usually carried out at
rock surface. It is difficult to estimate
three dimensional stress state. Elastic
constants are not required to estimate
rock stress.
Hydraulic fracturing method A borehole is drilled from the ground
surface or a roadway. Initial stress is
estimated from hydraulic fracturing data.
Only horizontal stresses are usually
estimated. It can be apllied up to
several km deep. There are any results.
Methods using oriented core Material tests in laboratory are carried out
for rock cores. Initial stress is estimated
from such data as stress-strain curves.
Results similar to other methods are
often obtained although the
mechanisms are not well understood.
Method based on fault
earthquake data
Orientation of initial stress is estimated
based on the focal mechanism of fault
earthquakes.
Enormous data can be used although
the stress magnitude can't be estimated.

Stress relief method
(1)
A borehole is drilled usually
from a roadway. The borehole
should be longer than the
roadway width to avoid areas
where stress concentrates.
A pilot hole is drilled from the
borehole top.
A probe is installed in the pilot
hole.
Overcoring is carried out
measuring deformation and/or
strains.
(2)
(3)
(4)
Stress relief methods

Principle of stress relief method
No stress exists in the hollow cylinder formed by
overcoring.
Magnitude of strain and/or deformation with overcoring
are equal to that when the pilot hole is drilled under the
initial stress state with an inverted sign.
The strains and/or deformations can be obtained by analytical
methods or numerical methods assuming an elastic medium.
Initial stress can be estimated by solving the simultaneous
equations.

Example
2
y
1
Overcoring
σ α = σ cos2
α+ σ sin 2
α + 2τ xycosα sinα
x y
x
3
A rosette gage was
atacched to a rock
surface and overcoring
was carried out around
the gage. Represent
change in strains of the
gauges 1, 2 and 3 by E,
ν, σx, σy τxy.
Conical bottom strain
method (Sakaguchi et al.,
1994)

Hydraulic fracturing method
Often used method.
Originally developed for
wells for petroleum and
geothermal energy to
measure stress and to
enhance the prodcution
Fracture
There are some cases in
which hydraulic fracturing
p
is carried out from an
existed roadway.
Packer
Hydraulic fracturing

A borehole is drilled.
Packer and a water pipe is
installed.
Water is injected measuring Fracture
pressure and flow rate.
p
Packer

Water valve is closed after
breakdown which is a decrease of
water pressure and represents that a
fracture appears at the borehole wall. Breakdown
The pressure at the breadown is
called pb.
Water is injected again.
Decrease of the slope of the water
pb
pr
Down-hole pressure
ps
pressure-time curve represent
reopening, namely, the fracture is
opened again. The pressure at re-
opening
pressure pr.
is called re-opening
Injection
and then
pressure
is continued for a while
the valve is shut.Water
Flow rate
will converge. The
converged pressure is called shut-in
pressure ps.
Time

Packer etc. are removed and
fracture orientation is
observed by a borehole
Breakdown
camera or impression packer. pb
pr
Down-hole pressure
ps
Flow rate
Time

Principle of hydraulic fracturing
Tangential stresses σA, σB at points A and B when internal
pressure p acts to a circular hole under maximum principal
stress σH and minimum principal
ignored for convenience).
σh
σ
stress (pore pressure is
H
A
σ A
σ
= 3σ h − σ H − p
σh
= 3σ −σ − pB H h p
B

principle of hydraulicfracturing
σA is smaller than σB. A fracture
initiate and grows from point A
when the following criterion
satisfied.
T0 is tensile strength.
is
σH
A
T0 ≤ −σ A = −3σ h + σ H + p
σh
The following equation is
derived for the breakdown
pressure pb.
T0 = −3σ h +σ H + pb
p
B

Principle of hydraulic Fracturing
Reopening occurs when the
tangential stress at point A
becomes tensile.
0 ≥ 3σ h − σ H − p
σH
A
Consequently, for reopening
pressure,
σh
0 = 3σ h − σ H − pr
p
B

Principle of hydraulic fracturing
It is said that shut-in
pressure is roughly
σh.
σ h = ps
equal to
σH
A
σh
p
B

Procedure to estimate initial stress
Minimum principal stress is estimated from the shut-in pressure
σ h = ps
Maximum principal stress is estimated from the minimum principal
stress and the re-opening pressure.
0 = 3σ h − σ H − pr
Orientation of the maximum principal stress is equal to that of the
fracture observed by a borehole camera or an impression packer.

Vertical fracture (left) and horizontal fracture
(right)

Criterion for horizontal fracture
Vertical stress σv at borehole wall is
σ v = σ V − 2 p
Criterion for horizontal fracture is
σ v < 3σ h − σ H
Another method has to be used in
fracture occurs.
he case where horizontal

More detail
Consideration on pore pressure and fracture mechanics
（Nihon Kikai Gakkai, 1989)
Determination of three dimensional stress state based on
data from one borehole (Nihon Kikai Gakkai, 1989)
Statistical consideration (Shin & Okubo, 1999)
Estimation of initial stress measuring strains (Sato et
1999b、Itoh et al., 2001)
Detailed consideration on re-opening pressure (Ito et
1999b)
al.,
al.,

Estimation of initial stress from laboratory test on oriented cores (Ex.
Nihon Kikai Gakkai, 1989)
AE mehod (Lavrov, 2003)
DSCA method (Oikawa et al.,1995, Yamaguchi et al., 1991, Matsuki et al.,
1995)
DRA mehod
ASR method
Mehod utilizing P-wave velocity
Results similar to such reliable methods as stress relief method and
hydraulic fracturing method are often obtained.
Principle is not well known. There are many points which should be
clarified. For example, how long rock core maintain the stress memory
is not well known.
Methods using oriented cores

Example of DRA method
b3 b3
6040
Loading30
40
20
20
10
0 0
0 0.1 0.2 0 10 20 30 40
Strain (10-2
) Stress (MPa)
Stress-strain curves Strain difference function
Stress(MPa)
Straindifference(10-6
)
Unloading

Sign and amplitude of elastic wave from fault slip depends
on orientation to the observatory.
Compressional and dilatational wave can be observed for
P-wave, for example.
Compressional wave
Dilatational wave
Method based on fault earthquakes

Compression and
tensile axes can be obtained by projecting polarity of P-wave on the
direction of observatory.
Directions of compression and tension axes are sometimes regarded as those
of the maximum and minimum principal stress, respectively.

Compression and tension axes represents stress change due
to fault slip and their directions should not be always
coincides to those of initial stress. However, similar results
to stress relief method and hydraulic fracturing method are
often obtained.
The directions of initial stress can be easily estimated from
enormous fault earthquake data although the magnitude of
initial stress can't be estimated.

Borehole breakout
Failure phenomena which are
observed at sidewall of
petroleum and geothermal wells
Failure zones of dog ear-shape
grow in the direction of the
minimum principal stress
Initial stress magntude can be
estimated from the shape of the
failure zone.
For detail, refer Brudy &
Zoback (1999), Cuss et al.
(2003), Haimson & Lee (2004)
σH
σh
Other methods

Core discing
Rock core breaks in a
many discs when a
borehole is drilled to a
high pressure zone.
Relationship between
stress state and disc
shape is investigated
(ex. Obara et al., 1998).
Initial stress state can
be roughly estimated. Core discing which was
observed at Kamaishi Mine

Other other other
Calcite twins （Kang et al., 1999）
Sub-crater of volcano (Karino and Murata, 1998）
Electric resistivity (Ito et al., 1999a）
etc.........................

Magnitude and orientation of Insitu stresses vary considerably within
geological systems.
2. The pre-existing stress state changes dramatically due to
excavation/construction therefore load must be redistributed.
3. Stress is not familiar – it is a tensor quantity and tensors are not
encountered in
everyday life.
4. It is a means to analyze mechanical behaviors of rock.
5. It serves as boundary conditions in rock engineering problems as a
stress state
is applied for analysis and design.
6. It helps in understanding groundwater fluid flow.
7. At large scale shed some light on the mechanism causing tectonic
plates to
move or fault to rupture with the added uncertainty in that there is no
constraint
on the total force, as is the case with gravity loads.

A section, normally less than 1m in length, of a borehole is sealed off with a straddle
packer.
The sealed-off section is then slowly pressurized with a fluid, usually water.
This generates tensile stresses at the borehole wall. Pressurization continues
until the borehole wall ruptures through tensile failure and a hydro fracture is
initiated.
The fracture plane is normally parallel to the borehole axis, and two fractures are
initiated simultaneously in diametrically opposite positions on the borehole
periphery.
The hydro fracture will initiate at the point, and propagate in the direction, offering
the least resistance. The fracture will therefore develop in a direction perpendicular
to the minimum principal stress.
The orientation of the fracture is obtained from the fracture traces on the borehole
wall –it coincides with the orientation of the maximum horizontal stress, in a vertical
or subvertical
hole where it is assumed that one principal stress is parallel to the borehole.
The fracture orientation may be determined either by use of an impression packer
and a compass or by use of geophysical methods such as a formation micro-scanner
or a borehole televiewer.

In its conventional form, the method is 2D: only the maximum and minimum normal
stresses in the plane perpendicular to the borehole axis are established.
For a vertical borehole, these components are the maximum and minimum horizontal
stresses.
Since the principal stress directions in tectonically passive and topographically at areas
areusually close to horizontal and vertical, it can often be assumed that the components
measured in a vertical borehole are two of the principal stresses.
Hydraulic fracturing is an efficient method for determining the 2D stress field, normally
inthe horizontal plane, and is therefore suitable at the early stages of projects when no
underground access exists.
Due to its efficiency, it is especially advantageous for measurements at great depth. . The
method is also not significantly affected by the drilling processes.
Hydraulic fracturing normally includes large equipment, which requires space.
Furthermore,the theoretical limitations normally imply that the measurements should
be done in vertical
holes. Hence, the method is most suited for surface measurements in vertical or
subvertical
boreholes.
Applied packer pressure – 2-4 MPa

Rock mechanics for engineering geology part 3

Rock mechanics for engineering geology part 3

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Rock mechanics for engineering geology part 3