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### Rock Strength

Maurice Dusseault

Common Symbols in RM

- E, n: Young’s modulus, Poisson’s ratio
- f, r, g : Porosity, density, unit weight
- c′, f′,To: Cohesion, friction , tensile strength
- T, p, po: Temperature, pressure, initial pres.
- UCS: Unconfined Compressive Str. (σ′3 = 0)
- sa,sr: Axial, radial stress (σ′ = eff. Stress)
- τ, σ1-σ3: Shear stress, deviatoric stress
- k: Permeability

These are the most common symbols we use

Chalk Fracture Experiments…

a. Three-point loading

c. Pure axial tension

2P

Surface of fracture

Surface of fracture

?

?

Compressive stress

T

T

Tensile stress

Fracture location indeterminate

P

P

Fracture location directly under 2P

P

P

b. Four-point loading

d. Pure torsion (moment)

Compressive stress

Surface of fracture

Surface of fracture is a helical shape

M

M

Tensile stress

Maximum tensile

stress orientation

Fracture location indeterminate, but between the two vertical loads

P

P

What is Strength?

- It turns out that strength of a geomaterial is a highly complex subject…
- There still remain issues on which the experts are not in full agreement (e.g. the scale dependency of tensile strength)
- 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.)

Yield Modes around a Borehole…

HMAX

High pw leads to slip of a joint or fault.

Usually occurs only with high MW

Brittle beam bucking under high uniaxial local loads

Crushing in breakout apex

Slip plane

Destabilized column is “popping” out

breakout

HMAX

Axial extension fractures: tensile failure when pw >

- 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 , perhaps arising from heating, low ʹ

High MW – Low MW

Shear failure in low cohesion rock precedes sloughing of cavings

Rock Strength

- Strength is the ability to sustain (support):
- Shear stress (shear strength)
- Compressive normal stress (crushing strength)
- Tensile stress (tensile strength)
- Bending (bending or beam strength)
- All of these depend on effective stresses (s), thus, we must know the pore pressure (p or po)
- Rock specimen strength is different than rock mass strength (joints, fissures, fissility…)
- Tests are done on cores, chips, analogues…

Rock Mass vs Sample Strength

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

slip

planes

a = 1

max planes

Field stresses

Z

r

r = 3

cores

Triaxial

Test

Stresses

a

faults

joints

fractures

grains

bedding

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

Sample damage can change properties!

How We Measure Rock Strength…

- Simple Measurements (no s3 – “index” tests)
- Uniaxial Compressive Strength (UCS), s3 = 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

n

rock

slip plane

Beam Bending Tests

Prepared specimen tests

P/2

P/2

Beam-bending

“tensile” test

P/2

P/2

Core tests

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

X-section

To

Effect of Flaws on ToP/2

P/2

Beam-bending

tensile test

P/2

P/2

Thus, To is flaw size dependent

Surface finish

A: notched

B: rough

C: polished

A

B

C

Surface 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

Brazilian indirect tension test, common usage

F

Nevertheless… Tensile 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.

To = F/A

F

Prepared rock specimen

Pure tension test:

extremely rare

A

F

Other 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

L

L

jack

round

flat

rock

epoxy

load

scriber

pull

slab

rebound

measure

mandril

core

Schmidt Hammer or Sclerometer

Recommendation

- Include a strength index test program in all your core analyses
- 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

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?

Core #2 ~27500′ below sea level

8000

Scratch test results

7000

6000

5000

Correlation to UCS (psi)

4000

This core section is quite homogeneous throughout

3000

2000

~ 1 foot (300 mm) length

1000

0

Red line is output from the load cell

Blue line is an averaged value of the force

Core #1 ~28000′ below sea level

Scratch test results

Heterogeneous and damaged core

Correlation to UCS (psi)

~ 1′ (300 mm)

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

Core #2 ~27500′ below sea level

Scratch test results

Heterogeneous, damaged core

Correlation to UCS (psi)

~ 1 foot (300 mm) length

How We Measure Rock Strength…

- Direct Measurements with confinement (s3)
- Compressive Strength: Triaxial Testing Machine
- Encase the core in an impermeable sleeve
- Confining stress is applied s3 first
- σa is then increased while…
- Pore pressure constant
- Record Δσa, εa, εr, ΔV
- More exotic tests…
- ΔT, Δp, even Δchemistry
- Creep tests (constant σ, measure )
- Hollow cylinders…

a = 1

slip

planes

max planes

r

r = 3

Triaxial

Test

Stresses

a

Strain rate

slip plane

rock

Rock Strength- Shear Strength- Shear strength: a vital geomechanics measure, used for design
- Shearing is associated with:

n

- 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

σ′3

σ′1

sn is normal effective stress

t is the shear stress,║to slip plane

σa

thin porous stainless steel cap for drainage

strain gauges

εa, εr

σr applied through oil pressure

load platen

seals

impermeable membrane

σa

The Triaxial CellCSIRO-Australia

Triaxial Test Apparatus

- A confining stress s3 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

Triaxial test cell. (

a

) Cell components including spherical seats

A

, end caps

B

, specimen

C

,

fluid port

D

, Strain Gauges

(optional), and polyurethane rubber sleeve

F

. (

b

) Insertion of

E

sleeve. (

c

) End cap removed to clean accumulated rock fragments. (

d

) Triaxial test with cell

in position, showing four-column load frame, 200-t hydraulic jack load cell and pressure

transducer connected to cell for automatic plotting of stress path.

(Franklin and Hoek, 1970.)

Sing06.049

f = 30% in situ, 35% in the core lab

damage

σ′a

σ′r

Shaley zone, sheared

Oil sand, intact

Some Issues in Testing- Sample disturbance
- Sample homogeneity
- Representativeness
- Specimen preparation
- Lateral e data? DV?
- Δp response (shales)?
- Testing must be done by a commercial lab with good QC
- Even then, results are critically examined

A σ′ ε Curve for a Rock Specimen

σ1 - σ3

massive damage,

shear plane develops

´3 =

´r

Y - peak

strength

damage

starts

sudden stress drop (brittle)

cohesion

breaking

stress difference

continued damage

ultimate or

residual

strength

“elastic” part of σ′ε curve

εa

seating, microcrack closure

axial strain

Shear Failure of Sandstone

a = 1

- High quality cylinder
- L = 2D
- Flat ends
- High angle shear plane
- Zone of dilation and crushing

r = 3

a

Yield and Strain-Weakening…

- At one σ′3 value…
- YP: peak shear strength
- YR: residual strength
- Red curve is at a high confining stress – s3
- Blue curve, low s3
- E.g.: high pore pressure = low s3, low strength, brittle behavior!
- Low po… higher strength, ductile

’ – e behavior

YP

ductile yield

YR

strain-weakening

Stress – sa - sr

YP

brittle rupture

YR

Axial strain - ea

Dilatancy During Rock Shearing

- When rocks shear under low s3, they dilate (+V)
- Dilation = +DV increase in pore V, microcracks
- Rock is weakened, but with higher f, k values
- For reservoirs and high p, s3 is low; dilation enhances permeability, an important factor in heavy oils…

s’ – e behavior

Y

Stress – sa - sr

Axial strain - ea

+DV

dilation = +ΔV

Axial strain - ea

-DV

compression

30

´3 = ´r

20

10

Full Triaxial Data Set Example´1 = ´a

´3 = 14.0 MPa

s-e curves

a- r (MPa)

´3 = 7.0 MPa

´3 = 2.0 MPa

UCS, ´3 = 0

Strain- %

0

+V

Strain- %

-V

1%

Volume change measurement (dilation)

Rock Strength: the M-C Plot

- To plot a yield criterion from triaxial tests, on equally scaled t, s axes, plot s1 and s3 at fail-ure, join with a semicircle, then the tangent = Y

t

- Y

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

cohesion

c

To

sn

s1

s3

Different s3 values

Triaxial s′-e Curves

This is extremely complex, and partly “machine dependent”…

s1 - s3

smax- peak

strength

massive damage,

shear plane develops

Y

damage

starts,

~0.60smax

sudden stress drop (brittle or strain-weakening)

cohesion

breaking

continued damage

E

stress difference

Yr

ultimate or

residual

strength

“elastic” part ofs-ecurve

specimen seating, microcrack closure

axial strain -ea

Behavioral Model Representation

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…

A

s-e curves

B

Deviatoric stress

E

D

C

E

+ve

DV (Deviatoric component)

A

B

C, D

-ve

Strain (%)

Geomaterial Failure Modes

-ve confinement (i.e. tension)

Low confinement

High confinement

- Tensile rupture
- Single plane
- Minimal general
- damage
- Fracture
- mechanics

- Shear rupture
- Bifurcation plane
- Some general damage
- Localization effects
- Plasticity theory
- with shearing

- General damage
- No bifurcation
- Distributed
- general damage
- Continuum damage
- mechanics best

Standard Approximation for Y

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

t- shear stress

Rocks are weak in tension!!

real Y

′

cohesion

Y = M-C Yield Criterion:

f = c + n·tan for n≥ To

f = 0 for n < To

c′

Y - linear

approximation

To

sn- normal stress

Use a Reasonable Y Approximation

Mohr-Coulomb failure (or yield) criterion

t- shear stress

real Y

standard

approximation

cohesion

Rocks are weak in tension: assume To = 0?

′

c′

This is a better approximation for cases of low confining stress

To

sn- normal stress

Different Methods to Present Y

- 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 s
- 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…

s1s3Plotting Method

- Plot s1, s3 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!)

s1

Curved or

linear fit

tan2a

s1 - s3·tan2a - Co = 0

Uniaxial

compressive

strength, Co

(not same as To)

s3

p q Plotting Method

- p = (s1 + s2)/2
- q = (s1 - s2)/2
- p is mean sn
- q is shear stress
- This method is most common in soil mechanics
- Sometimes used in petroleum rock mechanics

q

actual curve

a

straight line assumption

p

q = Co/2 + p·tana

Why Approximations for Y?

- 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

Still the Best (in my view)!

Mohr-Coulomb yield criterion

t - shear stress- σ1 – σ2

2

Y

tf

′a = ′1

σ′n

cohesion

′r = ′3

tf

shear planes

c’

sn- normal stress

σ’3

σ’1

UCS

To ~ 0.12·UCS

σ′n

What is “Failure”?

- 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)

sMAX

smin

Breakouts

Rock Strength – Sophisticated Tests

Radial section

- 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.

Axial section

Geomaterial Characteristics

- Strain-weakening (peak strength & residual strength, YP, YR, are different)
- failure strain, f = ƒ(3 ): f as 3
- YP, YR = ƒ(3 ): YP, YR as 3
- YP/YR = ƒ(3 ): YP/YR as 3
- Dilatancy
- +V with increasing 1 - 3
- dilatancy suppressed as 3
- Thus we must link stresses, strains and failure behavior in our rock testing

List of Tests* from Core Lab Ltd.- A

- 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

List of Tests from Core Lab Ltd. - B

- 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

σr = σ3

σr = σ3

σa = σ1

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…

Diagenesis and Strength

slip

planes

a = 1

max planes

diagenesis

effects on the

Mohr-Coulomb

strength envelope

shear stress

r

r = 3

chemical cementation

Triaxial

Test

Stresses

a

densification

(more interlock)

original

sediment

cohesion

diagenetic

strength

increase

c

normal

stress

s3

s1

Extreme Diagenesis Case…

Xal overgrowths, interpenetrative structure!

Rock Strength – Shear 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

Area - A

N - normal force

S

Shear box

S - shear

force

N

S

A

Linear “fit”

Curvilinear “fit”

data point

N

c

A

cohesion

Shearbox 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”

A

N

A

N

slip plane

S

rock

Yield 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 surface only, not the rock mass…

= t

Linear “fit”

Y

Curvilinear “fit”

data point

c

cohesion

= sn

Cohesion 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

1 - 3

complete s-e curve

cohesion

mobilization

stress difference

friction

mobilization

a

Friction

- Frictional resistance to slip between surfaces
- Must have movement () 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

mob = cohesion + friction f = c′ + n

This is merely the M-C curve

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

Crushing 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 )
- Tests involve increasing all-around effective stress (′) equally, measuring V/′
- Tests can involve reducing p in a highly stresses specimen (ie: ’ increases as p drops)

*UCSS = unconsolidated sandstone

High-Porosity Chalk

Hollow, weak grains (coccoliths)

Weak cementation (dog-tooth calcite)

Weak, cleavable grain mineral (CaCO3)

Cracks and Grain Contacts

- Point-to-point contacts are much more compliant than long (diagenetic) contacts
- Large open microfissures are compliant
- Oriented contacts or microfissures give rise to anisotropy of mechanical properties
- Rocks with depositional structure or exposed to differential stress fields over geological time develop anisotropy through diagenesis

Particle Contacts

- In granular media, macroscopic stresses are transmitted through grain contact forces (fn, fs)
- These particles may be crushable (coal, feldspars, coccoliths…)

fs = shear force

fn = normal force

A non-Crushing Rock

- 25% porosity SiO2 sandstone (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

Crushing Strength

- Apply p, ( > p), allow to equilibrate ( = - p)
- Increase by increasing or dropping p = - p
- Record ΔV/V, plot versus effective stress
- The curve is the crushing behavior with +

ΔV

p

LE

crushing

material

ΔV

Cracks and Grain Contacts

E2

E1

Microflaws can

close, open, or

slip as σ′ changes

E1

E3

- Flaws govern rock stiffness

The nature of the grain-to-grain

contacts and the overall porosity

govern the stiffness of porous SS

Griffith Flaw Theory

- Rock is a “flawed” material
- Flaws are stressed (t) by deviatoric loads
- Tensile s develops at tips
- Rock is weak in tension
- Tensile cracks form easily
- They propagate to 3
- As L, acceleration
- Sudden rupture ensues

induced tensile fracture

initial flaw

deviatoric loading

Low Confining Stress (Low σ′3)

- Low s′3: brittle rupture
- Fractures develop parallel to s′1
- Specimen may separate into several “columns”
- When column L/w ratio becomes large, buckling or crushing
- Rupture is sudden
- e energy is released

s′1

Edge

chipping

Axial fractures

L

s′3

Surface spalls

w

Yield at Higher s′3

Crystal debonding

Stress-induced anisotropy

q

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

Microfissure

fabric at yield

q

High s3

Intercrystalline forces at yield

How Do We “Test” This Rock Mass?

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

A large core specimen

A core “plug”

1 m

Machu Picchu, Peru, Inca Stonecraft

Lessons Learned

- Strength is a complex issue
- We have to use “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 the field and the lab is best…

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