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Rock Strength (Shear strength, Rock Strength (Shear strength, compressive strength, tensile strength, compressive strength, tensile strength,

fracture strength, crushing strength, etcfracture strength, crushing strength, etc……) )

Maurice Dusseault

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Common Symbols in RMCommon Symbols in RM

� E, ν: Young’s modulus, Poisson’s ratio� φ: Porosity (e.g. 0.25, or 25%)� c′, φ′,To: Cohesion, friction ∠, tensile strength� T, p, po: Temperature, pressure, initial pres.� σv, σh: Vertical and horizontal stress� σhmin, σHMAX : Smallest, largest horizontal σ� σ1,σ2,σ3:Major, intermediate, minor stress� ρ, γ: Density, unit weight (γ = ρ × g)� K, C: Bulk modulus, compressibilityThese are the most common symbols we use

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Chalk Fracture ExperimentsChalk Fracture Experiments……

a. Three-point loading 2P

P PTensile stress

Compressive stressSurface of fracture

Fracture location directly under 2P

b. Four-point

loading

P

P P

Tensile stress

Compressive stress Surface of fracture

Fracture location indeterminate, but between

the two vertical loads

Pd. Pure torsion (moment)

c. Pure axial tension

Maximum tensilestress orientation

M M

T T

Surface of fracture

Fracture location indeterminate

? ?

Surface of fracture is a helical shape

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What is Strength?What is Strength?

� It turns out that the “strength” of a geomaterial is a highly complex subject…

� There still remain issues on which experts are not in full agreement (e.g. scale dependency of tensile strength, “creep failure”…)

� Strength can be a simple measure, such as unconfined compressive strength (UCS)

� It can be complex (shear stress under a full 3-dimensional stress state, under temperature and elevated pore pressure conditions, etc.)

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Yield Modes around a BoreholeYield Modes around a Borehole……

σHMAX Brittle beam bucking under high uniaxial local loads

breakout

Destabilized column is “popping” out

High pw leads to slip of a joint or fault.

Usually occurs only with high MW Slip

plane

Axial extension fractures: tensile failure when pw > σθ

Shear failure in low cohesion rock precedes sloughing of cavings

σHMAX

High MW – Low MW

Crushing in breakout apex

Plus some others…

�Fissile shale sloughing if the borehole

axis is close to parallel to fissility axis

�Swelling, weakening, turning to “mush”

�Borehole surface spalling from high σθ,

perhaps arising from heating, low σʹθ

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Rock StrengthRock Strength

� Strength is the capacity to sustain (support):�Shear stress (shear strength)

�Compressive normal stress (crushing strength)

�Tensile stress (tensile strength)

�Bending stress (bending or beam strength)

� All of these depend on effective stresses (σ′), thus, we must know the pore pressure (p or po)

� Rock specimenstrength is different than rock massstrength (joints, fissures, fissility…)

� Tests are done on cores, chips, analogues…

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Rock Mass vs Sample StrengthRock Mass vs Sample Strength

faultsjoints

fracturesgrains

bedding

coresZ

Scale differences and flaws mean that direct extrapolation of test results to the field is difficult in

Petroleum Geomechanics

Field stresses

Sample damage can change properties!

Field scale: grains to kilometers. Lab scale: grains to 100 mm diameter specimen.

100 m

σa

σrσr = σ3

σa = σ1 τmax planesslip

planes

TriaxialTest

Stresses

100 mm

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How We Measure Rock StrengthHow We Measure Rock Strength……

� Simple measurements (called “index” tests)�Uniaxial Compressive Strength (UCS), σ′3 = 0

�Tensile strength (pure tension, hard for rocks!)

� Indirect tensile test (Brazilian test)

�Point-load, beam-bending, scratch test, needle…

� Direct shear tests of a planar surface�A joint surface, bedding plane, fault plane,

sheared plane, lithologic interface, etc.

�The surface is loaded, then sheared, and the resistance to shear loads is recorded

rock

σ′nτ

slip plane

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Beam Bending TestsBeam Bending Tests

P/2

P/2

Beam-bending

“tensile” testP/2

P/2

X-section

Core testsThe “strength” of a rock in beam bending is strongly dependent on the specimen size and the “quality” of the surface finish!

Prepared specimen tests

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Effect of Flaws on TEffect of Flaws on Too

P/2

P/2ƒ

ToA B C

A: notchedB: roughC: polished

Surface finishThus, To is flaw size dependent

Beam-bending

tensile test

P/2

P/2

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Surface Finish and FlawsSurface Finish and Flaws

� If the strength of rock in beam-bending depends on the surface finish…�Beam-bending is of little value to engineering

design issues

�The strength of rock in tension is dependent on the size of the flaw

� These observations, combined with the fact that rock masses have many discontinuities…�Suggest that tensile strength may be inappropriate

for many rock engineering problems

�Measurements of To are of questionable value

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Cracks and FlawsCracks and Flaws

P/2

P/2 P/2

P/2

Rock

Flaw

The flaw tip concentrates stresses, making brittle

rupture much easier

Stress “trajectories”

Flaws in Rock :•Pores•Grain boundaries•Joints•Fractures•Faults, etc.

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Stress ConcentrationsStress Concentrations

Hoek and Bieniawski 1965, Int J Frac Mech 193) 137-155

www.webpages.uidaho.edu/~simkat/geol542.html

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““ GriffithGriffith 11”” CracksCracks……

� Flaws serve as local stress concentrators

� The applied stress is far lower than the yield value for the “intact” material…

� Stress concentration at the flaw tip locally overcomes material strength�ƒ(crack length, orientation, sharpness)

� Strength of a “flawed” brittle material is therefore far lower than an “intact” material

� All rocks are flawed! Hence, weak in tension, strong in compression

1 - Griffith AA. 1920. The phenomena of rupture and flow in solids. Phil Tran Roy Soc of London A221:163-198

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NeverthelessNevertheless…… Tensile StrengthTensile Strength……

� Tensile strength (To) is extremely difficult to measure: it is direction-dependent, flaw-dependent, sample size-dependent, ...

� An indirect method, the Brazilian disk test, is used

� For a large reservoir, To may be assumed to be zero because of joints, bedding planes, etc.

A

To = F/APrepared

rock specimen

Pure tension test:extremely rare

F

F

Brazilian indirect tension test, common usage

F

F

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jack

L

slab

scriberpull

load epoxy

rock

flat

L

round

core

reboundmeasure

mandril

Schmidt Hammer or Sclerometer

Other Index Tests of StrengthOther Index Tests of Strength

� Index tests: strength “indices” only

� Point load tests on core disks or chunks

� Brinnell Hardness test, or penetration test

� Scratch test on slabbed core sections

� Sclerometer or steel bar rebound tests

� All these use correla-tion charts for strength estimates

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RecommendationRecommendation

� Include a strength index test program in all your core analyses (good for drilling…)

� Service companies can provide this systematically upon request

� The data bank can be linked to compaction potential, drillability, other factors…

� It may help identify particularly troublesome zones that could cause troubles

� Any systematically collected data will prove useful for correlations and engineering

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Cautions!Cautions!

� However, index values are estimates only!

� Scale is a problem!�What is the scale of interest to the problem?

� Is testing a drill chip a test of sufficient scale?

� Is it possible to index test fractured reservoirs?

� Geometry is an issue!�How is strength affected by a 2 mm thick shale

lamination in an otherwise intact sandstone?

� Is strength anisotropic?

�What is orientation w.r.t. the anisotropy?

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Core Core ##2 ~275002 ~27500′′ below sea levelbelow sea level

0

1000

2000

3000

4000

5000

6000

7000

8000

Cor

rela

tion

to U

CS

(ps

i)

~ 1 foot (300 mm) length

Scratch test results

Red line is output from the load cellBlue line is a moving average value of the force

This core section is quite homogeneous throughout

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0

1000

2000

3000

4000

5000

6000

27264.0 27264.1 27264.2 27264.3 27264.4 27264.5 27264.6 27264.7 27264.8 27264.9 27265.0

Depth (ft)

UC

S (

psi)

Cor

rela

tion

to U

CS

(ps

i)

~ 1′ (300 mm)

Scratch test results

Red line is output from the load cell. Blue line is an averaged value of the force

Core Core ##1 ~280001 ~28000′′ below sea levelbelow sea level

Heterogeneous and damaged core

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0

2000

4000

6000

8000

10000

12000

14000

16000

18000

27348.0 27348.1 27348.2 27348.3 27348.4 27348.5 27348.6 27348.7 27348.8 27348.9 27349.0

Depth (ft)

UC

S (

psi)

Epoxy Spike

Cor

rela

tion

to U

CS

(ps

i)

~ 1 foot (300 mm) length

Scratch test results

Core Core ##2 ~275002 ~27500′′ below sea levelbelow sea level

Heterogeneous, damaged core

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Rock StrengthRock Strength-- Shear StrengthShear Strength

� Shear strength: a vital geomechanics measure, used for design

� Shearing is associated with:•Borehole instabilities, breakouts, failure•Reservoir shear and induced seismicity•Casing shear and well collapse•Reactivation of old faults, creation of new ones•Hydraulic fracture in soft, weak reservoirs•Loss of cohesion and sand production•Bit penetration, particularly PCD bits

rock

σ′n

τslip planeσ′n is normal effective stress

τ is the shear stress,║to slip plane

σ′ 3

σ′1

σ′σ′σ′σ′n

ττττ

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How We Measure Rock StrengthHow We Measure Rock Strength……

� Direct measurements with confinement (σ′3)�Compressive Strength: Triaxial Testing Machine

�Encase the core in an impermeable sleeve

�Confining stress is applied σ3 first�σa is then increased while…

�Pore pressure constant

�Record ∆σa, εa, εr, ∆V

� More exotic tests…�∆T, ∆p, even ∆chemistry

�Creep tests (constant σ, measure ε)

�Hollow cylinders…

TriaxialTest

Stresses

Strain rate

σ′r = σ′3

σ′a = σ′1 σ′n

τf

shear planes

•

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High Confining Pressure FrameHigh Confining Pressure Frame

Core Lab Inc.

σ′a

σ′r

Press

Cel

l

Frame

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The Triaxial CellThe Triaxial Cell

load platen

p control, ∆Vp

εa, εr

impermeable membrane

σr ap

plie

d th

roug

h oi

l pre

ssur

e

strain gauges

σa

σa

seals

thin porous stainless steel

cap for drainage

CS

IRO

-Aus

tral

ia

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Triaxial Test ApparatusTriaxial Test Apparatus

� A confining stress σ′3 is applied (membrane)

� The pore pressure is zero, or constant

� The axial stress is increased slowly until well past peak strength

� Deformation behavior is measured during test

� T, po… can be varied in sophisticated systems

Sing06.049

Triaxial test cell. ( a) Cell components including spherical seats A, end caps B, specimen C,fluid port D, Strain Gauges E (optional), and polyurethane rubber sleeve F. (b) Insertion ofsleeve. ( c) End cap removed to clean accumulated rock fragments. ( d) Triaxial test with cellin position, showing four-column load frame, 200-t hydraulic jack load cell and pressuretransducer connected to cell for automatic plotting of stress path. (Franklin and Hoek, 1970.)

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φφφφ = 30% in situ, 35% in the core labdamage

Some Issues in TestingSome Issues in Testing

� Sample disturbance

� Sample homogeneity

� Representativeness

� Specimen preparation

� Lateral ε data? ∆V?

� ∆p response (shales)?

� Testing must be done by a commercial lab with good QC

� Even then, results are critically examined

Shaley zone, sheared

Oil sand, intact

σ′a

σ′r

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Shear Failure of SandstoneShear Failure of Sandstone

σa

σr = σ3

σa = σ1

� High quality cylinder

� L = 2D

� Flat ends

� High angle shear plane

� Zone of dilation and crushing

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A A σ′σ′ -- εε Curve for a Rock SpecimenCurve for a Rock Specimenst

ress

diff

eren

ceσ1 - σ3

axial strain

εa

Y - peakstrength

sudden stress drop (brittle)

massive damage, shear plane develops

continued damage

ultimate orresidualstrength

cohesionbreaking

damage starts

“elastic” part of σ′-ε curve

seating, microcrack closure

σ´3 = σ´r

σ′r = σ′3

σ′a = σ′1 σ′n

τf

shear planes

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Yield and StrainYield and Strain--WeakeningWeakening……

� At one σ′3 value…� YP: peak shear strength

� YR: residual strength

� Red curve is at a high confining stress –σ′3

� Blue curve, low σ′3� E.g.: high pore pressure

= low σ′3, low strength, brittle behavior!

� Low po… higher strength, ductile

Str

ess

–σσ σσ a

-σσ σσ r

Axial strain - εεεεa

strain-weakening

σ’ – ε behavior

YP

YR

brittle rupture

YP

YR

ductile yield

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Measurement of Measurement of ∆∆(Pore Volume)(Pore Volume)

� The specimen is isolated from all liquids in the cell

� The pore space is liquid-saturated (water or oil)

� As the axial stress is increased, ∆V measured

� This gives an estimate of the dilation of the rock as it shears

load platen

impermeable membrane

σr

= σ

3

σa

σa

seals

thin porous stainless steel

cap for drainage

σr

= σ

3

∆V

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Dilatancy During Rock ShearingDilatancy During Rock Shearing

� When rocks shear under low σ′3, they dilate (+V)

� Dilation = +∆V increase in pore V, microcracks

� Rock is weakened, but with higher φ, k values

� For reservoirs and high p, σ′3 is low; dilation enhances permeability, an important factor in heavy oils…

Str

ess

–σ a

-σ r

σ’ – ε behavior

+∆V

-∆V

compression

dilation = +∆V

Axial strain - εa

Axial strain - εa

Y

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Behavioral Model RepresentationBehavioral Model Representation

Strain (%)

Dev

iato

ric s

tres

s∆∆ ∆∆V

(D

evia

toric

co

mpo

nent

)

+ve

-ve

E

E

C, D AB

Constitutive models:A: Linear elastic, no deviatoric dilation

B: Perfect plasticity, no deviatoric dilation

C: Instantaneously strain-weakening, post failure dilation angle

D: Gradual weakening, post failure dilation

E: General response, simulated with more complex models…

σ−ε curves

A

B

C D

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Full Triaxial Data Set ExampleFull Triaxial Data Set Example

σ a-

σ r (

MP

a)

Strain- %0

1%

10

20

30

40

UCS, σ´3 = 0

σ´3 = 2.0 MPa

σ´3 = 14.0 MPa

Strain- %+∆V-∆V

Volume change measurement (dilation)

σ−ε curves

σ´3 = 7.0 MPa

1.0

-1.0

σ′r = σ′3

σ′a = σ′1 σ′n

τf

shear planes

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Rock Strength: the MRock Strength: the M--C PlotC Plot

� To plot a yield criterion from triaxial tests, on equally scaled τ, σ′ axes, plot σ′1 and σ′3 at fail-ure, join with a semicircle, then the tangent = Y

σ′3 σ′1To

σ′n

τ

cohesion

c

Different σ′3 values

4 triaxial tests are plotted here, plus the tensile strength from other tests

Y

Normal stress

Shear stress

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Standard Approximation for YStandard Approximation for Y

To σ′n - normal stress

τ - shear stress

Rocks are weak in tension!!

Y = M-C Yield Criterion:τf = c′ + σ′n·tanφ′ for σ′n ≥ To

τf = 0 for σ′n < To

real Y

Known as the Mohr-Coulomb (M-C) failure criterion

cohesion

Y - linearapproximation

φ′

c′

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Geomaterial Failure ModesGeomaterial Failure Modes

Shear rupture•Bifurcation plane•Some general damage•Localization effects•Plasticity theory

with shearing

Low confinement

General damage•No bifurcation•Distributed

general damage•Continuum damage

mechanics best

High confinement

Tensile rupture•Single plane•Minimal general

damage•Fracture

mechanics

-ve confinement (i.e. tension)

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Rock Failure Modes and Rock Failure Modes and σσ′′33

van der Pluijm and Marshak, 1997

Davis and Reynolds, 1996

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Use a Reasonable Use a Reasonable YY ApproximationApproximation

To σ′n - normal stress

τ - shear stress

cohesion

c′

standardapproximation

real Y

Mohr-Coulomb failure (or yield) criterion

φ′

This is a better approximation for cases of low confining stress

Rocks are weak in tension: can

we assume that To = 0 for large

scale problems?

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Different Methods to Present Different Methods to Present YY

� In the literature, there are > five different methods to plot rock yield (failure) criteria

� The next slides show other examples of M-C yield criterion plots

� I prefer the Standard Mohr plot…�Simple, clear, easy circle construction for σ

� There are also many yield criteria: Lade, Druker-Prager, Modified Von Mises…

� I suggest: stick with the M-C criterion, it is robust, well understood, direct…

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σσ′′11 ×× σσ′′33 Plotting MethodPlotting Method

� Plot σ′1, σ′3 values at peak strength on axes

� Fit a curve or a straight line to the data points

� y-intercept is Co or UCS

� (Note that circles to solve for stresses can only be used on a conventional M-C plot, not on this one!)

σσσσ′′′′1 - σσσσ′′′′3 ·tan2αααα - Co = 0

σ′1

σ′3

Uniaxialcompressivestrength, Co

(not same as To)

tan2α

Curved orlinear fit

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pp ×× qq Plotting MethodPlotting Method

� p = (σ′1 + σ′2)/2� q = (σ′1 - σ′2)/2� p is mean σ′n� q is shear stress

� This method is most common in soil mechanics

� Sometimes used in petroleum rock mechanics

p

q

α

q = Co/2 + p·tanα

actual curve

straight line assumption

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Why Approximations for Why Approximations for YY??

� Linear approximations of Y are easy to understand: cohesion plus friction effects

� When yield (failure) criteria are plotted from real lab data, there is a lot of scatter

� Numerical modeling is the only way to use the full curvilinear Y envelopes…

� Thus, a linear approximation allows:�Quick calculations and estimates

�A clearer picture of the physics

�But! Use the right stress range to improve results of simple calculations

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Still the Best (in my view)!Still the Best (in my view)!

σ’1To ~ 0.12·UCS

σ′n - normal stress

cohesion

c’

φ’

Y

Mohr-Coulomb yield criterionτ - shear stress- σ1 – σ22

UCSσ’3

σ′r = σ′3

σ′a = σ′1 σ′n

τf

shear planes

σ′n

τf

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What is What is ““ FailureFailure”” ??

� Be very careful!! Failure is a loss of function. Rock yield is a loss of strength.

� Don’t confuse them!!

� For example:� Most boreholes have

“yielded” zones

� But, the hole has not collapsed! …or “failed”!

� It still fulfills its function (allowing drill advance)

σMAX

σmin

Breakouts

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Rock Strength Rock Strength –– Sophisticated TestsSophisticated Tests

� More sophisticated rock tests are also possible and useful in some circumstances:

� Hollow cylinders, cavity tests…

� Creep tests for salt and soft shales

� Thermal effects tests

� Shale properties with different geochemistry

� Drillability tests… etc.

Radial section

Axial section

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Shear of Weak SandShear of Weak Sand

Cou

rtes

t Nor

weg

ian

Geo

tech

nica

l Ins

titut

e

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Geomaterial CharacteristicsGeomaterial Characteristics

� Strain-weakening (peak strength & residual strength, YP, YR, are different)� failure strain, εf = ƒ(σ′3 ): εf ↑ as σ′3↑�YP, YR = ƒ(σ′3 ): YP and YR ↑ as σ′3 ↑�YP/YR = ƒ(σ′3 ): YP/YR ↓ as σ′3 ↑

� Dilatancy �+∆V ↑ with increasing σ1 - σ3

�dilatancy suppressed as σ′3 ↑� Thus we must link stresses, strains and failure

behavior in our rock testing

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List of Tests* from Core Lab Ltd.List of Tests* from Core Lab Ltd.-- AA

� Triaxial and uniaxial testing for E, ν, UCS � Sonic velocity testing for dynamic Young's

Modulus and Poisson's Ratio (ED, νD)� Mohr-Coulomb envelope analysis and

construction � Sonic velocity & acoustic impedance under σ′

� Sonic velocity in crude oils and brines � Seismic velocity testing at 10Hz � Hydrostatic and uniaxial pore volume

compressibility � Calibration of sonic and dipole log

*Downloaded list from their website

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List of Tests from Core Lab Ltd. List of Tests from Core Lab Ltd. -- BB

� Fracture azimuth and max stress azimuth (sonic velocity anisotropy)

� Evaluation of natural fracture conductivity � Thick wall cylinder test (sand production onset)� Fracture toughness analysis � Brinell hardness test for closure stress analysis � Proppant embedment testing vs. closure stress � Brazilian indirect tensile strength � Point load tensile test for wellbore stability

analysis www.corelab .com/PetroleumServices/Advanced/RockMech.asp

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Strain and Shear Slip (Displacement)Strain and Shear Slip (Displacement)

� As σa increased, εa took place as well

� However, when yield took place, deformation only along slip plane!

� We cannot speak about strain any more, only slip or deformation

� Also, there is elastic strain and plastic strain…

σa = σ1

σa = σ1

σr=

σ3

σr=

σ3

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Diagenesis and StrengthDiagenesis and Strength

shear stress

normalstress

chemical cementation

densification(more interlock)

originalsediment

diageneticstrengthincrease

σ′1σ′3

cohesion

diagenesiseffects on the

Mohr-Coulomb strength envelope

σa

σrσr = σ′3

σa = σ′1 τmax planesslip

planes

TriaxialTest

Stresses

c′

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Extreme Diagenesis CaseExtreme Diagenesis Case……

Xal overgrowths, interpenetrative structure!

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Rock Strength Rock Strength –– Shear Box TestsShear Box Tests

� Strength of joints or faults require shear box tests

� Specimen must be available and aligned properly in a shear box

� Different stress values (N) are used

N - normal force

S - shearforceN

S Shear box

Area - A

SA

NA

Linear “fit”

Curvilinear “fit”

data point

cohesionc

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Shearbox ApparatusShearbox Apparatus

� Normal load (stress)

� Reaction frame with high stiffness

� Split box for rock specimens to be tested � Different sizes can be

mounted in the device

� Shear displacement on lower half of “box”

� Rails allow continuous “self-centering”

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Yield Criterion in Shearbox TestsYield Criterion in Shearbox Tests

� Also a Mohr-Coulomb plot

•The points representing normal load and shear load are plotted on axes of equal

•As with triaxial test data, a straight line fit is often used

•This Y is for the joint surface only, not for the entire rock mass…

SA

NA

Linear “fit”

Curvilinear “fit”data point

cohesionc

= τ

= σ′n

Y

rock

N

Sslip plane

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Cohesion and FrictionCohesion and Friction……

� Bonded grains

� Crystal strength

� Interlocking grains

� Cohesive strength builds up rapidly with strain

� But! Permanently lost with fabric damage and debonding of grains

stre

ss d

iffer

ence

σ1 - σ3

εa

complete σ-ε curve

cohesion mobilization

frictionmobilization

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FrictionFriction

� Frictional resistance to slip between surfaces

� Must have movement (ε) to mobilize it

� Slip of microfissures can contribute

� Slip of grains at their contacts develops

� Friction is not destroyed by strain and damage

� Friction is affected by normal effective stress

� Friction builds up more slowly with strain

τmob = cohesion + friction τf = c′ + µσ′nThis is merely the M-C curve

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Estimating Rock StrengthEstimating Rock Strength

� Laboratory tests OK in some cases (salt, clay), and are useful as indicators in all cases

� Problems of fissures and discontinuities

� Problems of anisotropy (eg: fissility planes)

� Often, a reasonable guess, tempered with data, is adequate, but not always

� Size of the structure (eg: well or reservoir) is a factor, particularly in jointed strata - scale

� Strength is a vital factor, but often it is difficult to choose the “right” strength value

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Strength AnisotropyStrength Anisotropy

Vertical core

Bedding inclination

θθ

0° 30° 60° 90°

UCS

θ

UCS

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Crushing StrengthCrushing Strength

� Some materials (North Sea Chalk, coal, diatomite, high porosity UCSS*) can crush

� Crushing is collapse of pores, crushing of grains, under isotropic stress (minimal τ)

� Tests involve increasing all-around effective stress (σ′) equally, measuring ∆V/∆σ′

� Tests can involve reducing p in a highly stresses specimen (ie: σ’ increases as p drops)

*UCSS = unconsolidated sandstone

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HighHigh--Porosity ChalkPorosity Chalk

Hollow, weak grains (coccoliths)

Weak, cleavable grain mineral (CaCO3)

Weak cementation (dog-tooth calcite)

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Cracks, Grain Contacts, StrengthCracks, Grain Contacts, Strength

� Point-to-point contacts are much weaker than long (diagenetic) contacts , lower Y

� Large open microfissures also lead to lower strength (cracks = stress concentrators)

� Oriented contacts or microfissures give rise to anisotropy of rock strength as well

� Rocks with anisotropic fabric or exposed to differential stress fields over geological time develop strength anisotropy as well

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Particle Contacts & StrengthParticle Contacts & Strength

� In granular media, contact forces(fn, fs) govern strength

� fs]f = c′ + µ⋅fn

� Again, the Mohr-Coulomb frictional concept…

fs = shear force

fn = normal force

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A NonA Non--Crushing RockCrushing Rock

� 25% porosity SiO2sandstone (99.5% Q)

� Contacts are diagenetic in nature…

� This gives a very high stiffness, even though…

� The rock is totally devoid of cohesion!

� Athabasca oil sands, Faja del Orinoco sands, are “similar” to this

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Crushing StrengthCrushing Strength

� Apply p, σ (σ > p), allow to equilibrate (σ′ = σ - p)

� Increase σ′ by increasing σ or dropping p

∆σ′ = ∆σ - ∆p

� Record ∆V/V, plot versus effective stress

� The curve is the crushing behavior with +∆σ′

σ

σ

σ

σ

p

∆V

LE

crushingmaterial

σ′

∆V

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σ′nnormal stress

cohesion

c’

Mohr-Coulomb yield criterionτ - shear stress- σ1 – σ22

σ′r = σ′3

σ′a = σ′1 σ′n

τf shear planes

Initial state

σ′1σ′3

shear yield, th

en collapseFabric collapse

Collapsing in a ChalkCollapsing in a Chalk

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cohesion

c’

Mohr-Coulomb yield criterionτ - shear stress- σ1 – σ22

σ′r = σ′3

σ′a = σ′1 σ′n

τf shear planes

Initial state

shear yield, th

en collapseFabric collapse

σ′nnormal stress

Increased effective stress leading to pore collapse

Increased pore pressure leading to shear, then collapse

Stress Changes Leading to CollapseStress Changes Leading to Collapse

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Griffith Flaw TheoryGriffith Flaw Theory

� Rock is a “flawed” material

� Flaws are stressed (τ) by deviatoric loads

� Tensile σθ develops at tips

� Rock is weak in tension

� Tensile cracks form easily

� They propagate ⊥ to σ3

� As L↑, acceleration

� Sudden rupture ensues

initial flaw

induced tensile fracture

deviatoric loading

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Low Confining Stress (Low Low Confining Stress (Low σ′σ′33))

� Low σ′3: brittle rupture

� Fractures develop parallel to σ′1

� Specimen may separate into several “columns”

� When column L/w ratio becomes large, buckling or crushing

� Rupture is sudden

� ε energy is released

Axial fractures

Edgechipping

Surface spalls

σ′1

σ′3

w

L

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θ

θ

Microfissurefabric at yield

Intercrystalline forces at yield

Stress-induced anisotropy

At high stress, general damage occurs; much microfissuring, then coalescence takes place (“bifurcation”)

High σ3

Crystal debonding

Yield at Higher Yield at Higher σσ′′33

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Low Low σσ33, Jointing Dominates, Jointing Dominates

σ1

σ1

σ3

σ3

fractures

blocksloosened

initial jointing

This becomes important in fractured shale drilling, in reservoir mechanics of jointed weak limestones…

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How Do We How Do We ““ TestTest”” This Rock Mass?This Rock Mass?

� Joints and fractures can be at scales of mm to several meters

� Large φ core: 115 mm

� Core plugs: 20-35 mm

� If joints dominate, small-scale core tests are “indicators” only

� This issue of “scale”enters into all Petroleum Geomechanics analyses

1 m

A large core specimenA core “plug”

Machu Picchu, Peru, Inca Stonecraft

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Lessons LearnedLessons Learned

� Strength is a complex issue

� We develop behavior “models” to understand strength and to apply it in practice

� Rock mass vs. rock specimen strength problem

� Problems of scale, heterogeneity…

� Nevertheless, rock strength can be measured and used to predict slip, crushing, triggering of fault movement, and so on.

� Combination of field and lab data is best…