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

Initial stress

Stress state near opening isdisturbedDisturbed zone is avoided for measurement of initial stressDisturbed 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 verticalequal to theoverburden(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 13.5MPa

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 ratiok 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 verticalstress is roughly equal to the primitive estimation

Initial horizontalstress 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 horizontalstress = depth independent value + depth proportional value for 0.25 - 0.33 ofpoisson's ratio

400

600

800

10000 10 20 30 40 50 60

Stress (MPa)

Dep

th (m

)

σV

Depth independent value?

Movement of the tectonic platesSpherical shell subsidence model

"Reprinted from http://pubs.usgs.gov/gip/earthq1/fig1.gif with permission from USGS".

Orientation of the maximum horizontal stress measured byhydraulic fracture method (Goodman, 1980)

Average focalmechanism 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 bygravity force. Compressive strainappears by the sphericalgeometry.

Need for initial stress measurementInitial stress can roughly estimated by the primitivemethodMeasurement 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 usuallyfrom a roadway. The boreholeshould be longer than theroadway width to avoid areaswhere 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 byovercoring.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

2y

1

Overcoring

σ α = σ cos 2 α + σ sin 2 α + 2τ xy cosα sin αx y

x

3A 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 strainmethod (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 inwhich hydraulic fracturing

pis carried out from anexisted roadway.

Packer

Hydraulic fracturing

Hydraulic fracturing

A borehole is drilled.Packer and a water pipe isinstalled.Water is injected measuring Fracture

pressure and flow rate.

p

Packer

Hydraulic fracturingWater valve is closed afterbreakdown which is a decrease ofwater pressure and represents that afracture appears at the borehole wall.

BreakdownThe pressure at the breadown iscalled pb.Water is injected again.

Decrease of the slope of the water

pb

pr

Down-hole pressureps

pressure-time curve representreopening, namely, the fracture isopened again. The pressure at re-openingpressure pr.

is called re-opening

Injectionand then pressure

is continued for a whilethe valve is shut.Water

Flow rate

will converge. Theconverged pressure is called shut-inpressure ps.

Time

Hydraulic fracturingPacker etc. are removed and fracture orientation is observed by a borehole Breakdown

camera or impression packer. pb

pr

Down-hole pressureps

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 principalstress σH and minimum principal ignored for convenience).

σh

σ

stress (pore pressure isH

A

σ A

σ

= 3σ h − σ H − p

σh= 3σ − σ − pB H h p

B

principle of hydraulic fracturing

σA is smaller than σB. A fracture initiate and grows from point Awhen the following criterion satisfied.

T0 is tensile strength.

is

σH

AT0 ≤ −σ A = −3σ h + σ H + p

σhThe following equation is derived for the breakdownpressure pb.

T0 = −3σ h + σ H + pb

p

B

Principle of hydraulic Fracturing

Reopening occurs when thetangential stress at point Abecomes tensile.

0 ≥ 3σ h − σ H − pσH

AConsequently, for reopeningpressure,

σh

0 = 3σ h − σ H − prp

B

Principle of hydraulic fracturing

It is said that shut-inpressure is roughlyσh.

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 et1999b 、 Itoh et al., 2001)

Detailed consideration on re-opening pressure (Ito et1999b)

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 mehodASR methodMehod 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

Loading3040

20

2010

0 00 0.1 0.2 0 10 20 30 40

Strain (10-2) Stress (MPa)

Stress-strain curves Strain difference function

Stre

ss (

MP

a)

Stra

in d

iffer

ence

(10

-6)

Unloading

Sign and amplitude of elastic wave from fault slip dependson orientation to the observatory.Compressional and dilatational wave can be observed forP-wave, for example.

Compressional wave

Dilatational wave

Method based on fault earthquakes

Compression andtensile 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 dueto 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 areobserved at sidewall of petroleum and geothermal wellsFailure zones of dog ear-shape grow in the direction of the minimum principal stressInitial 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 amany 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 wasobserved 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 withingeological systems.2. The pre-existing stress state changes dramatically due toexcavation/construction therefore load must be redistributed.3. Stress is not familiar – it is a tensor quantity and tensors are not encountered ineveryday 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 stateis 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 tomove or fault to rupture with the added uncertainty in that there is no constrainton the total force, as is the case with gravity loads.

Insitu Stress

A section, normally less than 1m in length, of a borehole is sealed off with a straddlepacker.The sealed-off section is then slowly pressurized with a fluid, usually water.This generates tensile stresses at the borehole wall. Pressurization continuesuntil 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 areinitiated 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 subverticalhole 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 normalstresses in the plane perpendicular to the borehole axis are established.For a vertical borehole, these components are the maximum and minimum horizontalstresses.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 componentsmeasured 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 nounderground access exists.Due to its efficiency, it is especially advantageous for measurements at great depth. . Themethod 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 verticalholes. Hence, the method is most suited for surface measurements in vertical or subverticalboreholes.Applied packer pressure – 2-4 MPa

Jyoti KhatiwadaMsc , Engineering Geology.

Roll no. 10

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