1. Rock Strength
Maurice Dusseault

2. 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

3. Chalk Fracture Experiments…
a. Three-point loading
c. Pure axial tension 2P
Surface of fracture
Surface of fracture
Compressive stress ? ? T T
Tensile stress P P
Fracture location indeterminate
Fracture location directly under 2P
b. Four-point loading P P
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

4. 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.)

5. 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

6. 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…

7. Rock Mass vs Sample Strength
Field scale: grains to kilometers. Lab scale: grains to 100 mm diameter specimen.
a = 1
slip
planes
max planes
Field stresses Z
r = 3 r
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!

8. How We Measure Rock Strength…
Simple Measurements (no s3 – “index” tests)
Uniaxial Compressive Strength (UCS), s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
n 
rock
slip plane

9. Prepared specimen tests
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

10. Thus, To is flaw size dependent
Effect of Flaws on To
P/2
P/2
Beam-bending
tensile test
P/2
P/2  To
Thus, To is flaw size dependent
Surface finish
A: notched
B: rough
C: polished A B C

11. 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

12. 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
Brazilian indirect tension test, common usage F

13. 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
load
epoxy
pull
scriber
slab
rebound
measure
mandril
core
Schmidt Hammer or Sclerometer

14. 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

15. 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?

16. 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
Red line is output from the load cell
Blue line is an averaged value of the force

17. 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

18. Core #2 ~27500′ below sea level
Scratch test results
Heterogeneous, damaged core
Correlation to UCS (psi)
~ 1 foot (300 mm) length

19. How We Measure Rock Strength…
Direct Measurements with confinement (s3)
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 = 3 r
Triaxial
Test
Stresses a
Strain rate

20. 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
rock
n 
slip plane
sn is normal effective stress
t is the shear stress,║to slip plane

21. High Confining Pressure Frame
Core Lab Inc.
Frame
σ′a
σ′r
Cell
Press

22. The Triaxial Cell εa, εr σa σr applied through oil pressure
load platen
p control, ΔVp
εa, εr
impermeable membrane
σr applied through oil pressure
strain gauges σa
seals
thin porous stainless steel cap for drainage
CSIRO-Australia

23. Triaxial Test Apparatus
A confining stress s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
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 of
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

24. 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
f = 30% in situ, 35% in the core lab
damage
Shaley zone, sheared
Oil sand, intact
σ′a
σ′r

25. A σ′ ­ ε Curve for a Rock Specimen
σ1 - σ3
´1 = ´a
massive damage,
shear plane develops
Y - peak
strength
´3 =
´r
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

26. Shear Failure of Sandstone
High quality cylinder
L = 2D
Flat ends
High angle shear plane
Zone of dilation and crushing
r = 3 a

27. Yield and Strain-Weakening…
At one σ′3 value…
YP: peak shear strength
YR: residual strength
Red curve is at a high confining stress – s3
Blue curve, low s3
E.g.: high pore pressure = low s3, 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

28. Dilatancy During Rock Shearing
When rocks shear under low s3, they dilate (+V)
Dilation = +DV increase in pore V, microcracks
Rock is weakened, but with higher f, k values
For reservoirs and high p, s3 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

29. Full Triaxial Data Set Example10 20 30 40
´3 = 14.0 MPa
s-e curves
a- r ( MPa)
´3 = 7.0 MPa
´3 = 2.0 MPa
UCS, ´3 = 0
Strain- %
+V
Strain- %
-V 1%
Volume change measurement (dilation)

30. Rock Strength: the M-C Plot
To plot a yield criterion from triaxial tests, on equally scaled t, s axes, plot s1 and s3 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
sn
s3
s1
Different s3 values

31. Triaxial s′-e Curves
This is extremely complex, and partly “machine dependent”…
s1 - s3
smax- peak
strength Y
massive damage,
shear plane develops
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 of s-e curve
specimen seating, microcrack closure
axial strain -ea

32. 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 D E C
+ve E
DV (Deviatoric component)
C, D A B
-ve
Strain (%)

33. 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

34. 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
sn - normal stress

35. 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
sn - normal stress

36. 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…

37. s1  s3 Plotting Method
Plot s1, s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!)
s1
Curved or
linear fit
tan2a
s1 - s3 ·tan2a - Co = 0
Uniaxial
compressive
strength, Co
(not same as To)
s3

38. straight line assumption
p  q Plotting Method
p = (s1 + s2)/2
q = (s1 - s2)/2
p is mean sn
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·tan a

39. 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

40. Still the Best (in my view)!
t - shear stress- σ1 – σ2
Mohr-Coulomb yield criterion 2 Y ’ tf
′a = ′1
σ′n
cohesion
′r = ′3 tf
shear planes c’
sn - normal stress
σ’3
σ’1
UCS
To ~ 0.12·UCS
σ′n

41. What is “Failure”? sMAX smin
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

42. 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

43. 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

44. 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

45. 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

46. 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…

47. Diagenesis and Strength
slip
planes
max planes
diagenesis
effects on the
Mohr-Coulomb
strength envelope
shear stress
r = 3 r
chemical cementation
Triaxial
Test
Stresses a
densification
(more interlock)
original
sediment
cohesion
diagenetic
strength
increase c
normal
stress
s3
s1

48. Extreme Diagenesis Case…
Xal overgrowths, interpenetrative structure!

49. 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
cohesion A

50. 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”

51. Yield Criterion in Shearbox Tests
Also a Mohr-Coulomb plot
rock N S
slip plane
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… S A
= t
Linear “fit” Y
Curvilinear “fit”
data point c
cohesion N A
= sn

52. 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

53. This is merely the M-C curve
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

54. 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

55. Strength Anisotropy  UCS UCS Vertical core 0° 30° 60° 90°  
Bedding inclination

56. 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

57. High-Porosity Chalk Hollow, weak grains (coccoliths)
Weak cementation (dog-tooth calcite)
Weak, cleavable grain mineral (CaCO3)

58. 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

59. 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

60. 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

61. 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

62. Cracks and Grain ContactsE1 E2
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

63. 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

64. 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

65. 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

66. Low s3, Jointing Dominates
blocks
loosened
initial jointing
fractures s3 s1

67. 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: 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

68. 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…