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# Compressive strength of rocks

Mechanical failure in rocks generally means either fracturing or permanent deformation as a result of compression. While many methods for calculating failure relationships exist, an initial measure of the compressive strength of reservoir rocks is still needed for use in those calculations.

## Estimating compressive strength

General rock failure criterion can be reduced to a few parameters dependent on lithology (m) and the uniaxial compressive strength (C0). Lithology is commonly derived during log analysis, so m may be estimated (Table 1)[1]. What is needed still is an initial measure of rock strength provided by C0. C0 can be estimated from porosity or sonic velocities, but many factors such as grain size, clay content, or saturation have significant influences.

A large amount of C0 data is available and, although there is considerable scatter, C0 usually varies systematically with other rock characteristics. We will concentrate on porosity as the primary controlling factor because it is routinely available from logs and is a fundamental input into reservoir simulators.

### Relating compressive strength to porosity

Numerous relationships have been developed to estimate C0, often in conjunction with general rock strength relationships. Table 2 lists many of the proposed relations for C0, some of which are plotted for various rock types in Fig. 1 and for sandstones in Fig. 2. We expect C0 to decrease as porosity increases. At some transition porosity, rocks will lose all initial strength and become merely a loose aggregate. No matter which relationship is chosen, variables such as cementation, alteration, texture, and so on can cause significant scatter.

....................(1)

....................(2)

If we accept the restrictive relationships for failure of Eq. 1 or 2, we can derive C0 from any such strength data:

....................(3)

### Critical porosity

However, this equation predicts a finite strength even as porosity approaches 1.0. More realistic forms must be used so that strength vanishes at some porosity Φc. This limiting porosity was shown as a crossover porosity from rock to a slurry by Raymer et al.[2] and was referred to as "critical porosity" elsewhere. Jizba[3] used such a concept to derive a general strength relationship for sandstones:

....................(4)

where τ and σn are the shear and normal stresses at failure.

The 0.36 within the parentheses is her presumed value for Φc. Notice, however, that this form indicates that sandstones have no tensile or cohesive strength. We can obtain a better result by using Jizba’s relationship at elevated confining pressure (say, 50 MPa), where it is more valid, and recasting the trend in terms of 'Eq. 2, as we did for the Scott relation.[4] Dobereiner and DeFreitas[5] measured several weak sandstones, and their results suggest that critical porosity is approximately 0.42. Using this critical porosity, we derive a uniaxial compressive strength

....................(5)

This C0 equation is plotted in Fig. 2 along with the modified Scott[6] and Jizba[3] equations and data of Dobereiner and DeFreitas.[5]

## Compaction strength

After a threshold region the rock begins to show ductile deformation under confining pressure. The pressure under which a rock deforms is known as its compaction strength.

In Fig. 3, at some elevated stress or confining pressure, the rock will begin to show ductile deformation. The grain structure begins to collapse, and the rock will compact and lose porosity. This compaction strength, Cc, is itself a function of porosity as well as mineralogy, diagenesis, and texture. In Figs. 4a and 4b, the behavior of two rocks under hydrostatic pressure is shown. The high-porosity (33%) sandstone (Fig. 4a) has a low "crush" strength of about 55 MPa. With a lower porosity of 19%, Berea sandstone has a much higher strength of 440 MPa (Fig. 4b). Notice that in both Figs. 4a and 4b, permanent deformation remains even after the stress is released. This hysteresis demonstrates the damage to the matrix structure caused by exceeding the crush strength.

In the cases in which studies are restricted to sandstones, an exponential dependence on porosity is usually observed (Fig. 4a). Scott[6] fit his and the Dunn et al.[7] data to the form

....................(6)

### Determining rock failure envelopes

With a general relationship available for uniaxial compressive strength and the compaction limit, rock failure envelopes can be determined for sandstones at any porosity. Fig. 5 shows the complete envelopes for the porosity range 0.15 to 0.35.

## Effect of clay content

Most sandstones are mixtures of mineral such as feldspars, calcite, dolomite, micas, clays, and of course quartz. Clays are a very common component and can make up anywhere from 0 to nearly 100% of a clastic rock. Usually, at some point greater than 40% clay, the rock is considered a shale or mudstone rather than a sandstone (refer to Gamma ray as a tool for rock characterization). The structure of clay minerals and their typical methods of bonding are significantly different from those of quartz, so we would expect clays to strongly influence mechanical properties. Such influences depend on the nature of the clay, the amount and location within the rock framework, and the state of hydration.

### Effect of clays on the mechanical properties of rocks

There have been few systematic studies of clay effects on the mechanical properties of rocks. Corbett et al.[8] demonstrated how the coefficient of internal friction and thus the strength of Austin chalk strongly depends on even a small clay fraction (Fig. 6).

In particular, smectite content was found to have more influence in this case than other clays. This allows us to derive a general relationship between failure and clay content.

....................(7)

where C is the smectite fraction. Unfortunately, this equation was developed for dry samples.

Jizba[9] tested several dry clay-rich samples and proposed a general linear envelope form for shales and shaley sandstones.

....................(8)

More relevant data, however, comes from Steiger and Leung[10] with both dry and saturated shale measurements (Fig. 7).

From these data, we derive an approximation for the wet shale uniaxial compressional strength.

....................(9)

This relation, as well as those for the Austin chalk, suggests a strong clay dependence. Jizba,[3] however, reported only a slight dependence of C0 on clay content in shaley sands.

It is likely that in many sands, clays reside as pore-filling materials and have only a secondary effect on mechanical properties. At this point, we expect clays to have a significant effect even in fairly pure sands (this will be seen also in sonic velocity measurements). Thus, a more general form for uniaxial compressive strength of sandstones would be

....................(10)

where the coefficient a has a value of approximately 100. The influence of clays on the mechanical properties of rocks needs much further investigation.

## Effect of pore fluids

Fluids can alter rock mechanical properties through fluid pressure, chemical reactions with mineral surfaces, and by lubricating moving surfaces. These effects are discussed in Pore fluid effects on rock mechanics.

## Effect of grain size and texture

In granular rocks, grain size also influences strength. For constant porosity, mineralogy, and texture, a smaller grain size means greater strength. This tendency has been observed in several sandstones and can be understood in terms of grain contact models.

Nelson[11] presents data on Navajo sandstone strength indicating a strong dependence on grain size. If a rock can be considered an aggregate of uniform spheres, smaller spheres will have more grain contacts per unit volume. Loads are distributed over more contracts, and each grain experiences lower stresses. Zhang[12] used Hertzian contact theory to calculate critical crushing strengths of quartz sands and found that porosity and grain radius combine to determine strength (Fig. 8)[13].

By fixing grain size, Zhang’s relationships could also provide crushing or compaction limits for sands at various porosities. For a grain size of 0.2 mm, we get a crushing strength, Cc, of

....................(11)

However, factors such as cementation and grain angularity will strongly alter this simple relationship.

If grains become cemented, not only does porosity decrease, but the effective area of intergranular contracts increases. Even small amounts of cement will increase strength substantially. Angularity of grains and sorting will also influence strength. More angular grains result in sharper point contacts, stress concentrations, and lower strength.

In general, if grain size is known to be smaller or cementation greater (for a given porosity and composition), then increased strength can be estimated by reducing the Hoek-Brown coefficient m. A value of m = 0 for siltstones and shales was suggested by Hoek and Brown.[14] Notice that this leads to minor contradiction because clays, with very fine grain size, weaken rocks. It is possible that many of Hoek and Brown’s "shales" were well indurated (slightly metamorphosed?), and grain size and increased cementation account for the increased strength (and reduced m). In rocks with low levels of diagenesis, clays reduce strength and require an increased m.

## Nomenclature

 C0 = uniaxial or unconfined compressive strength, GPa or MPa E = Young’s modulus, GPa or MPa G = shear modulus, GPa or MPa C = clay content m = Hoek-Brown strength coefficient σf = failure stress, GPa or MPa σn = normal stress, GPa or MPa Vp = compressional velocity, m/s Vs = shear velocity, m/s α = failure envelope slope ρ = density, kg/m3 or g/cm3 Φ = porosity τ = relaxation time, s–1 (radians/s)

## References

1. Hoek, E. and Brown, E.T. 1982. Underground Excavations in Rock. Amsterdam, The Netherlands: Elsevier Applied Science.
2. Raymer, L.L., Hunt, E.R., and Gardner, J.S. 1980. An Improved Sonic Transit Time-To-Porosity Transform. Presented at the SPWLA 21st Annual Logging Symposium, Lafayette, Louisiana, USA, 8–11 July. Paper P.
3. Jizba, D.L. 1991. Mechanical and acoustical properties of sandstones and shales. PhD dissertation, Stanford University, Palo Alto, California.
4. Scott, T.E. 1989. The effects of porosity on the mechanics of faulting in sandstones. PhD dissertation, University of Texas at Dallas, Dallas, Texas.
5. Dobereiner, L. and De Freitas, M.H. 1986. Geotechnical properties of weak sandstones. Geotechnique 36 (1): 79-94. http://dx.doi.org/10.1680/geot.1986.36.1.79.
6. Kazi, A., Sen, Z., and Sadagah, B.-E.H. 1983. Relationship Between Sonic Pulse Velocity And Uniaxial Compressive Strength Of Rocks. Presented at the 24th U.S. Symposium on Rock Mechanics (USRMS), College Station, Texas, USA, 20-23 June. ARMA-83-0409.
7. Dunn, D.E., LaFountain, L.J., and Jackson, R.E. 1973. Porosity dependence and mechanism of brittle fracture in sandstones. J. Geophys. Res. 78 (14): 2403-2417. http://dx.doi.org/10.1029/JB078i014p02403.
8. Corbett, K., Friedman, M., and Spang, J. 1987. Fracture Development and Mechanical Stratigraphy of Austin Chalk, Texas. AAPG Bull. 71 (1): 17-28. http://dx.doi.org/10.1306%2F94886D35-1704-11D7-8645000102C1865D.
9. Dowla, N., Hayatdavoudi, A., Ghalambor, A. et al. 1990. Laboratory investigation of saturation effect on mechanical properties of rocks. Presented at the 31st SPWLA Annual Logging Symposium, Lafayette, Louisiana, USA, 24–27 June. Paper EE.
10. Steiger, R.P. and Leung, P.K. 1989. Predictions of wellbore stability in shale formations at great depth. In Rock at Great Depth: Rock Mechanics and Rock Physics at Great Depth—Proceedings of an International Symposium, Pau, 28–31 August 1989, ed. V. Maury and D. Fourmaintraux, Vol. 3, 1209. London: Taylor & Francis.
11. Nelson, R.A. 1983. Geological Analysis of Naturally Fractured Reservoirs. Houston, Texas: Gulf Publishing Company.
12. Zhang, J.-X. 1991. Mechanical compaction and the brittle ductile transition in porous rocks: Geological implications for accretionary wedge seismicity. PhD dissertation, Stony Brook University, Stony Brook, New York
13. Morgenstern, N.R. and Eigenbrod, K.D. 1974. Classification of Argillaceous Soils and Rocks. Journal of the Geotechnical Engineering Division (ASCE) 100 (10): 1137-1156.
14. Hoek, E. and Brown, E.T. 1988. The Hoek-Brown Failure Criterion—A 1988 Update. Proc., 15th Canadian Rock Mechanics Symposium, Toronto, Ontario, Canada, 31–38.

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