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.

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

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

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!

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

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

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 ~ 0.2-0.3 -For shales, n ~ 0.3-0.4 DL L strain (e) = L Δr n = ΔL

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′

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”

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…

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

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

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

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)

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.

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

Reminder: Scale and Heterogenity

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

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

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

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)

Reminder: Scale Issues… 70-200 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

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.

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?)

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 (<10-8-10-7), 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

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)

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!

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

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

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

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 

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.

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

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′

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 …)

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

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

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.

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

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

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)

“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

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