1. Rock and Rock Mass Deformability (Compressibility, Stiffness…)
Maurice Dusseault
Module 1-E: Deformability
The basic rock properties definitions: Young’s modulus and Poisson’s Ratio, compressibility. Measuring rock properties from core. The triaxial test and how it works. Measuring one-dimensional compressibility. The concept of different stress paths in the laboratory to emulate the different stress paths in the field. Estimating rock properties using indirect measures and correlations, such as geophysical logs, correlations to other measures such as porosity and seismic velocities, scratch tests, point load tests, and so on.

2. Common Symbols in RM E, n: Young’s modulus, Poisson’s ratio
f: Porosity (e.g. 0.25, or 25%)
c′, f′,To: Cohesion, friction , tensile strength
T, p, po: Temperature, pressure, initial pres.
sv, sh: Vertical and horizontal stress
shmin, sHMAX: Smallest, largest horizontal σ
s1,s2,s3: Major, intermediate, minor stress
r, g: Density, unit weight ( =  × g)
K, C: Bulk modulus, compressibility
These are the most common symbols we use

3. Obtaining Rock Deformability
Properties data bank
Depth Fric Coh. XXX YYY ZZZ
Commercial, in-house data banks
Log data
REG. TIPO
SVS-337
Rock Properties
(E, ν, , c′, C, k,…)
Reflected
and direct
paths
Borehole
seismics
Lab tests
We use a number of sources to get rock material properties. These include lab tets, core examination and quick tests, seismic data log data, correlations to data banks, and so on.
3-D Seismics
Core data

4. Stress and Pressure
Petroleum geomechanics deals with stress & pressure
Effective stress: “solid stress”
Pressure is in the fluid phase
To assess the effects of Δσ', Δp, ΔT, ΔC…
Rock properties are needed
Deformation properties…
Fluid transport properties…
Thermal properties…
sa – axial
stress
pore pressure A po
sr – radial
stress
ΔC refers to a change in concentration, such as a change in salinity in the pore liquid. For rocks such as quartzose sandstone and fractured carbonates, these changes have no significant effect on volume changes, stresses or pressures. However, in fine-grained rocks such as shale, a change in the concentration of a specie, such as a change from a Na-dominated pore liquid to a Ca-dominated pore liquid, can lead to a significant volume change. This, of course, leads to stress and pressure changes!

5. Rock Stiffness, Deformation
To solve a s-e problem, we must know how the rock deforms (strain - e) in response to Ds
This is often referred to as the “stiffness” (or compliance, or elasticity, or compressibility…)
For “linear elastic” rock, only two parameters are needed: Young’s modulus, E, and Poisson’s ratio, n (see example, next slide)
For more complicated cases - plasticity, dilation, anisotropic rock, salt, etc. - more and different parameters are needed

6. The Linear Elastic Model
The stiffness is assumed to be constant (E)
When loads are removed, deformation are reversed
Suitable for metals, low  rocks such as…
Anhydrite, carbonates, granite, cemented sands…
For many petroleum geomechanics problems, linear elastic assump-tions are sufficient
E1 is stiffer than E2
σ1 = σ′a σ3 E1
Stress – Δσ′a - (= σ1 – σ3) E2
Strain - εa

7. Definitions of E and ν? Ds Triaxial Test ε Δσ′ E = Δr n = ΔL
Young’s modulus (E): E is how much a material compresses under a uniaxial change in effective stress - Δσ′
Triaxial Test
deformation
Ds DL
Δσ′
E = ε
radial
dilation Dr
Poisson’s ratio (n): n is how much rock expands laterally when compressed. If n = 0, no expansion (e.g.: a sponge).
-For sandstones, n ~
-For shales, n ~ DL L
strain (e) = L Δr
n = ΔL

8. 1D and 3D Compressibility?
Change in volume with a change in stress
In 1-D compressibility, lateral strain, εh, = 0
Often used for flat-strata compaction analysis
3-D compressibility involves all-around Δσ′
C3D = 1/K, where K is the bulk modulus of elasticity
Δσ′v
cylindrical
specimen
εh = 0
εh = 0
Be very careful how compressibility is defined. It is the deformation associated with a change in effective stress at a constant pore pressure. It IS NOT the deformation associated with a change in pore pressure.
Δσ′
Δσ′
Δσ′
Δσ′

9. Some Guidelines for Testing…
Use high quality core, as “undisturbed” as possible, under the circumstances…
Avoid freezing, other severe treatments
Preserve RM specimens on the rig floor if you can
Use as large a specimen as possible
A large specimen is more representative
Avoid “plugging” if possible (more disturbance)
If undisturbed core is unavailable
Analogues may be used
Data banks can be queried
Disturbed samples may be tested with “judgment”

10. More Guidelines for Testing…
Replicating in situ conditions of T, p, [σ] is “best practice” (but not always necessary)
Following the stress path that the rock exper-iences during exploitation is “best practice”
Test “representative” specimens of the GMU
Testing jointed rock masses in the laboratory is not feasible; only the “matrix” of the blocks…
It is best to combine laboratory test data with log data, seismic data, geological models, and update the data base as new data arrive…

11. Laboratory Stiffness of Rocks
From cores, other samples: however, these may be microfissured (Efield may be underestimated)
In microfissured or porous rock, crack closure, slip, contact deformation may dominate stiffness
ES and nS (static) under s′3 gives best values
If joints are common in situ, they may dominate rock response, but are hard to test in the lab
Remember that in the lab, we may not be testing any joints at larger scale. In the field, the compressibility of the joint systems are vital in fractured rocks.
DT, Ds
in the lab
in situ
~0.10 m
10 m
Dp, DC

12. Typical Test Configuration…
An “undamaged”, homogeneous rock interval is selected
A cylinder is prepared with flat parallel ends
The cylinder is jacketed
Confining stress & pore pressure are applied
The axial stress is increased gradually
εa, εr (εr) measured σa D σr σr
L ~ 2D

13. Jacketed Cylindrical Rock Specimen
Strain gauges measure strains (or other special devices can be used)
Pore pressure can be controlled, and…
ΔVpore can be measured at constant backpressure
Similar set-up for high- T tests and creep tests
Acoustic wave end caps
Etc… σa
p control, ΔVp
thin porous stainless steel cap for drainage
strain gauges
εa, εr
σr applied through oil pressure
load platen
seals
impermeable membrane σa

14. A Simpler Standard Triaxial Cell
Developed by Evert Hoek & John Franklin
Is a good basic cell for rock testing
Standard test methods are published by the International Society for Rock Mechanics (ISRM)

15. In the Laboratory…
Axial deformation is measured directly by the movement of the test platens
Bonded strain gauges on the specimen sides are also used
Gives axial strain (calculate E)
Also gives the lateral strain (calculate )
Special methods for porous rocks or shales because strain gauges don’t work well
High T tests, acoustic velocity measurements during tests, etc., etc.

16. Reminder: Heterogeneity!
This is an example of the issue of homogeneity and a layered rock mass. Here we have a mixture of salt, a viscous substance, and anhydrite, a strong, low-porosity calcium sulfate rock. A high confining stress was used, and the salt behaved in a viscous manner, flowing outward radially under the high deviatoric stress (axial - radial stress). However, the anhydrite fractured with vertical tension cracks during this process, and neither flowed nor failed through shear or collapse. Interpreting such tests is extremely challenging, and so is the mathematical modelling of such a thinly layered sequence of materials of dramatically differing properties. These issues are still beyond or at the limits or our abilities in rock mechanics.
These materials respond radically different to stress: one flows, the other fractures. How might we incorporate such behavior in our testing and modeling for a natural gas storage cavern?
Original specimen Post-test appearance

17. Reminder: Scale and Heterogenity

18. Reminder: Anisotropy Different directional stiffness is common!
Bedding planes
Oriented minerals (clays usually)
Oriented microcracks, joints, fissures…
Close alternation of thin beds of different inherent stiffness (laminated or schistose)
Imbricated grains
Different stresses = anisotropic response
Anisotropic grain contact fabric, etc.
stiffer
less stiff

19. Apparent axial stiffness - M
Reminder: Anisotropy
Δσ′a
Apparent axial stiffness - M L
Vertical core  0°
30°
60°
90°  
Bedding inclination 0°
30°
60°
90°
e.g.: shales, laminated strata

20. Orthotropic Stiffness Model
In some cases, it is best to use an orthotropic stiffness model - shale
Vertical stiffness and Poisson’s ratio are different than the horizontal ones
Properties in the hori-zontal plane the same
This is as complex as we want: E1, E2, ν
Layered or laminated sedimentary strata σv σh
Strata subjected to different stresses σh
We rarely use models other than isotropic elastic and perhaps orthotropic.
Shales (clay minerals) E1
Orthotropic elastic model E2 ν

21. Lab Data, Then What?
Clearly, laboratory tests are valuable, but insufficient for design and optimization…
We also use correlations from geophysical logs
Obtain relevant, high-quality log data
Calibrate using lab test data
Use logs and 3-D seismic to extrapolate and interpolate (generating a 3-D whole earth model)

22. Reminder: Scale Issues…mm
Laboratory specimen (“intact”)
A tunnel in a rock mass
Rock vs Rock mass
--Intact rock
--Single discontinuities
--Two discontinuities
--Several disc.
--Rockmass
20-30 m

23. Rock Mass Stiffness Determination
Use correlations based on geology, density, porosity, lithology…
Use seismic velocities (vP, vS) for an upper-bound limit (invariably an overestimate)
Measurements on specimens in the lab? (problems of scale and joints)
In situ measurements
Back-analysis using monitoring data such as compaction measurements…
Reservoir response to earth tides…
Some methods will give underestimates of compressibility (overestimates of stiffness), often by a large amount. Seismic velocities can only be used reliably with good regional correlations and understanding of the lithology. Reservoir response to earth tides gives a global reservoir compressibility, but is considered to give too high a stiffness (response to small stress changes often underestimates compressibility at large stress changes). In Situ measurements and monitoring are likely the best, but the trouble is that economic and surface facilities decisions have to be made before significant monitoring data are available, especially in offshore cases.

24. In Situ Stiffness Measurements
Pressurization of a packer-isolated zone, with measurement of radial deformation (Δr/Δσ′), in an “impermeable material” so that Δσ′ = Δpw
Direct borehole jack methods (mining only)
Geotechnical pressure-meter modified for high pressures (membrane inflated at high pressure, radial deformation measured)
Hydrofracture flexing (THE™ tool, rarely used and quite expensive)
Correlations (penetration, indentation, others?)

25. Seismic Wave Stiffness (ED, nD)
vP, vS: dynamic responses are affected by stress, density and elastic properties (s, r, E, n)
Seismic strains are tiny (< ), they do not compress microcracks, pores, or contacts
Thus, ED is always higher than the static test moduli, ES
The more microfissures, pores, point contacts, the more ED > ES, x 1.3 to x 10 (for UCSS)
If porosity ~ 0, s’ very high, ES approaches ED
Seismic moduli should be calibrated by testing

26. Seismic (Dynamic) Parameters
DtS
P-wave (compressional wave)
Dt is transit time, plotted in micro-seconds per foot or per metre
Vp and Vs are calculated from the transit time and the distance – L – between the receiver and the transmitter in the acoustic sonde
DtP
Amplitude
S-wave (shear wave)
time to tp ts
Vp = L/Dtp
Vs = L/Dts
Dynamic Elastic Parameters:
D = [Vp2 - 2Vs2]/[2(Vp2 - Vs2)]
ED = [b.Vs2(3Vp2 - 4Vs2)]/(Vp2 - Vs2)
D = b.Vs2 (shear modulus)

27. Deformation Properties from Logs
Simple P-wave transit time correlations
Dipole sonic data - Vp and Vs
Often dipole sonic data are not available
Estimate Vs from Vp w known ratios, lithology…
Basic data needed for Vs estimation:
Sonic log, preferably dipole
Density log (gamma-gamma)
Water saturation log (for corrections)
Mineralogy/lithology logs (for corrections)
Service companies provide these methods
Use with JUDGMENT!

28. Multiple Receiver Sonic Log
Acoustic sonde, multiple receivers
disturbed zone
Damage will alter the sonic velocities.
Wave trains
Velocity curve is offset because of lower v near the borehole wall (damage)
Attentuation per metre can also be used to relate to damage
Arrival time delay from damage
location
constant velocity, intact rock
borehole wall
receivers
ray
paths
source
time
lower velocity region

29. Back-Analysis for Stiffness
Apply a known effective stress change, measure deformations (eg: uplift, compaction)
Use a mathematical model to back-calculate the rock properties (best-fit approach)
Includes all large-scale effects
Can be confounded by heterogeneity, anisotropy, poor choice of GMU, ...
Often used as a check of assumptions
One must commit to some monitoring (e.g. {Δz}) in order to achieve such results

30. Discontinuities & E
Grain contact deformability is responsible for sandstone stiffness
These may be cemented or not, and in low- media, they become interlocked, rocks are stiffer
fn = normal force

31. Cracks and Grain Contacts
Microflaws can
close or open as s changes E1 E2 E1 E3
Flaws govern rock stiffness
Cracks between grains, as well as larger-scale flaws such as joint planes, are responsible for much of the deformability of materials such as fractures shales and limestones. The stiffness is a function of how open the cracks are, what is the crack density, how long they are…
The contact fabric and  dominate the stiffness of porous SS
Fissures are more important in limestones, as well as 

32. Grain Contact Stiffness
A grain contact solution was developed 120 years ago by Hertz
Δd  1/E, (ΔF)⅔
It shows that grain-to-grain contacts become stiffer with higher load
High  rocks dominated by such contacts
They are stiffer with stress: C = ƒ(σ′)
d - Δd ΔF
Δσ′v
Hertz
“Reality”
The top is the Hertz contact stiffness model which is used to explain why unconsolidated and high porosity sandstones have a non-linear stiffness. The rocks are stiffer under increased confining stress because the grain contacts increase in area, therefore reducing the amount of deformation for a unit increase in contact force.
In practice, we cannot calculate stiffness from the fabric, so we take measurements in appropriately designed test devices, in this case a 1-D loading device.

33. Non-Linear Elastic Behavior
The stiffness is assumed to be variable – E(σ′3)
Deformation is still reversible
Suitable for highly microfissured materials, high  granular rocks
For some geomechanics problems, a non-linear elastic solution is useful
Sand compaction, sand production… σ1 σ3 E1
Stress – (σ1 – σ3)
A deformation curve for a non-linear elastic material. The rock is stiffer with increased stress, which usually accompanies depletion. Thus, in depleted sandstone reservoirs, we find that the rock stiffness will increase substantially, enough so that it is easily detected on seismic surveys. Conversely, if we use a high-pressure waterflood, there are reductions in seismic velocity and greater wave amplitude attenuation in the more highly pressured regions.
Carbonates also show similar effects, as well as attenuation effects of the shear wave amplitude in intensely fractured cases.
UCSS
crack closure
E2 = ƒ(σ′)
Strain - εa

34. Empirical relationships
Real Rock Behavior
Unconsolidated sandstones have a stiffness that is a function of effective stress - σ′:
Empirical relationships {
For modeling and calculations, we often use empirical stiffness laws based on tests or reservoir monitoring. These relationships usually have no formal basis in theory, although the power-law case can be “rationalized” in terms of Hertz theory. We consider these deformation “laws” as being correlative in nature rather than reflecting any underlying physics. Of course, we have to do this all the time in engineering, and it does not imply that this is an inferior approach. E
Linear elastic (E = constant) σ′

35. Geological Factors and Stiffness
Geological history can help us infer the stiffness and the response to loading…
Intense diagenesis
Reduces porosity
Cementation
Deeper burial then erosion (precompaction)
Age (in general correlated to stiffness)
Porosity (lower , higher E)
Mineralogy (SiO2 vs. litharenite mineralogy)
Tectonic loading (reduces …)

36. Sandstone Stiffness & Diagenesis
Dl = Df
low s, large A
sij
T, t, s, p
chemistry p
high s, small A
DIAGENESIS!

37. Precompaction Effect by Erosion
porosity
apparent threshold Ds
“virgin”
compression curve
present state
stiff reload
response
Always remember that the geological burial and tectonic history affects the stiffness of the rocks. For example, the quartzose sands of the Orinoco Oil Sands in Venezuela have are more compressible (by a factor of about 2 to 4) than the Athabasca Oil Sands in Alberta. This is in large part because the Alberta sediments are more than twice as old and were buried perhaps twice as deep before erosion too place.
log(sv)
The sand is far stiffer than expected because of precompaction! - e.g. Athabasca Oil Sand

38. High-Porosity Chalk Hollow, weak grains (coccoliths)
Weak cementation (dog-tooth calcite)
Weak, cleavable grain mineral (CaCO3)
Hollow grains or easily crushed grains such as shale fragments behave quite differently than intact quartz and feldspar grains. Not only is the compaction behavior radically different, the deformations are largely irreversible. Once the cementation breaks down, the grains begin crushing, and so on, the compaction arising from increases in stress is usually highly significant. In Lake Maracaibo, the shallow heavy oil sands that compact massively are not only young, they have never experienced high erosion, and the proportion of weak grains is far larger than in cases such as the oil sands of the North Slope of Alaska, Alberta, etc.

39. Cementation and Compaction
porosity
apparent threshold Ds’
collapse when grain cement ruptured
normal
densification
cementation effect
“virgin”
compression curve
initial stiff
response
In some cases, we see a sharp onset in compaction once some threshold is exceeded. This is the onset of the rupture of the fabric, and is most common in coal, North Sea Chalk. In high porosity litharenites, the compressibility is high and the deformation largely irreversible, but a sharp “corner” in the compressibility curve is not common.
log(s’v)
North Sea Chalk, high  Coal, Diatomite, are quasi-stable, collapsing rocks at some σ′v

40. Fine-grained unconsolidated sandstone
Sandstone Diagenesis
Dense grain packing
Many long contacts
Concavo-convex grain contacts
SiO2 precipitated in interstitial regions
Only 1% solution at contacts = 8% loss in volume
-A stable interpenetrative fabric, high stiffness
Fine-grained unconsolidated sandstone

41. Effect of Diagenetic Densification
porosity
apparent threshold Ds’
diagenetic
porosity loss
@ constant s’
“virgin”
compression curve
present state
stiff load
response
This is the conceptual model of the effect of diagenesis on stiffness. It also leads to an apparent threshold stress for triggering of larger non-recoverable components of deformation.
log(s’v)

42. “Precompaction” Effect
A threshold value is necessary before any non-elastic compression is triggered
This may arise from three processes
True precompaction by burial then erosion
Cementation of grains = stiffer + stronger
Prolonged diagenesis increases stifeness
Little deformation is seen in early drawdown, but occurs later
This can confuse field planning

43. Issues to Remember…
Stiffness (elastic modulus) is a fundamentally important rock property for analysis
We can measure it with cored rock specimens
Also, in boreholes (much more rarely)
Sometimes, through correlations to other measures such as geophysical data
Sometimes, through back-calculation, using deformation measurements
Nevertheless, there is always uncertainty
And, natural lithological heterogeneity