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Rock failure relationships
Understanding rock failure relationships is important because under reservoir pressure and stress conditions, production can induce rock failure, sometime with catastrophic effects. By applying strength criteria, within reservoir simulators we can predict when problems might occur.
Rock failure definition
Stress strain relationships in rocks examined the elastic behavior of rocks, which was largely reversible. Here we deal with permanent deformation. By rock failure, we mean the formation of faults and fracture planes, crushing, and relative motion of individual mineral grains and cements.
Failure can involve formation of discrete fracture zones and the more "ductile" or homogeneous deformation. The latter deformation is caused by a broad distribution of fracture zones or general grain crushing during compaction. We will not consider deformation caused by plastic strains of the mineral components, as is common in salt and in calcite at higher temperatures. In our analysis, several assumptions are made:
- Material is isotropic and homogeneous
- Stresses are applied uniformly
- Textural characteristics such as grain size and sorting have no influence
- Temperature and strain rate are ignored
- Intermediate stresses are presumed to play no role.
Each of these assumptions can be violated, and some have been demonstrated to have major influences on rock strength.
To begin with, a brief review of the standard Mohr failure criteria will be examined to introduce concepts and define terms, as well as to establish the basic mathematics behind the strength relationships.
Units of stress and strength are the same as pressure and are covered in Stress strain relationships in rocks. More detailed descriptions can be found in standard textbooks (i.e., Jaeger^{[1]}).
Mohr circles and linear failure envelope
Mohr circles and a linear failure envelope are the most common methods used to plot stresses and indicate strength limits. This technique predicts failure when stresses surpass both the intrinsic strength of a rock and internal friction.
The primary terms and characteristics are shown in Fig. 1.
- Normal stresses across any plane are plotted on the horizontal axis, and shear stresses are plotted on the vertical axis.
- Compressive stresses are defined as positive (as opposed to the mechanical engineering convention of tensional stresses being positive).
- For the hydrostatic case, all stresses are equal; this stress state is represented by a point on the horizontal axis.
- When stresses differ, the maximum principal stress, σ_{1}, and minimum stress, σ_{3}, are plotted on the horizontal axis and the possible shear stresses along any plane fall on a hemisphere connecting σ_{1} and σ_{3} (Fig. 2).
The mean stress, σ_{m}, and radius of this circle, r, are simple sums and differences of the principal stresses.
The normal stress across any plane, σ_{n}, and the shear stress along the plane, τ, are functions of the principal stresses and the plane orientation.
where θ is the angle between the plane and the σ_{3} direction.
From Eqs. 5 and 6, the maximum shear occurs along a plane oriented at (45°). However, because of friction, rocks do not fail along this plane. Instead, failure occurs along some rotated plane where friction is lower, yet shear stress is still high. This failure point (or plane) is shown in Fig. 1 as the nearly diagonal line. Fig. 1 also shows the associated normal and shear stresses. If numerous failure tests are made and plotted, an envelope is defined as in Fig. 2. In this case, friction is assumed to be a simple linear function of normal stress, and the resulting envelope is also linear. The slope of this envelope is α, and we define μ as the angle of internal friction
Within this framework, we can define several important properties of the rock as shown in Fig. 2: C_{0} = Uniaxial or unconfined compressive strength (σ_{3} = 0) C_{u} = Cohesive strength or the intercept of the envelope with σ_{n} = 0 C_{t} = Tensional strength
The failure envelope is then defined by the line
If the rock has already been broken, or a fracture already exists, then both C_{u} and C_{t} will be close to zero.
Several useful equations can be derived from the geometric relationships shown so far. From the equation for a circle,
At the intersection of the envelope and the circle, we must have
which leads to
Using the general solution to a second-order polynomial gives
Because we want only a point where the circle touches the envelope, the square root term must vanish.
After some algebraic manipulation, we find
and
Substitution of C_{u} (defined in Eq.14) into Eq. 12 gives an expression for normal stress.
If the envelope could be continued into the tensional region, the tensional strength could easily be obtained:
Under tension, the stresses are negative, although the tensional strength is a positive number. Thus, if rocks could fail according to a constant internal friction, we would have a simple way to relate the stresses involved and need only a couple of material constants, such as C_{u} and α.
Limitation of Mohr's approach
We are immediately faced with two problems when we try to apply Coulomb-Navier failure criteria:
- Rocks do not generally have a linear failure envelope
- Material properties controlling failure must be obtained either through logs or assumed behavior.
Fig. 3 shows the type of envelope commonly seen. In fact, we know that the slope must change as stresses are increased because rocks begin yielding and act more plastically. Fig. 4 shows the generalized behavior expected. At normal stresses above the brittle-ductile transition, failure can no longer be maintained on a single plane, but is distributed more homogeneously throughout the sample. We must develop different failure criteria, one that produced an appropriately curved envelope, and we expect it to have a strong porosity dependence (Fig. 5^{[2]}).
Other failure approaches
Numerous failure criteria have been proposed that are primarily empirically based. Table 1 shows some of the criteria proposed both for general purposes and for specific rock types or conditions. Observed failure envelopes are smooth forms so simple exponential or power-law functions can usually be found that fit the data well. The relations of Bienlawski^{[3]} and Hoek and Brown^{[4]} are most common. Much of the recent work in rock mechanics has been directed toward ascertaining the constants of these relationships in terms of easily measurable rock properties. Note that these relationships apply primarily to the brittle failure regime and cannot be used for grain crushing or pore collapse (as discussed in Compressive strength of rocks) or when substantial ductile or plastic deformation is involved. We will examine these proposed forms to interrelate terms and reduce unknowns to variables that can be derived from logs.
Hoek-Brown equations
Hoek and Brown^{[4]} compiled extensive data on a variety of rock types and produced relationships that are simple and can be developed into forms amenable to well-log analysis. A primary feature of this failure criterion is a relation between the maximum and minimum stresses when both are normalized by the uniaxial compressive strength
This formulation was motivated by the systematic behavior seen in many tests as shown in Fig. 6. InEq. 18,mand sare material constants dependent on the overall quality of the rock mass, and mis also dependent on the rock type ('Table 2).
Note that we could derive the value for m from a mineralogic analysis. In our analysis, we will presume that the local rock mass of interest is intact, and thus
For applications that are in sandstones, numeric results can often use
Eq. 18 can be rewritten to give one principal stress in terms of the other:
Such normalized stress states were used to construct the curved envelope in Fig. 4.
The tensional strength, the stress at which an envelope would cross the horizontal axis, is found by equating σ_{1} to σ_{3} in Eq. 21 (note that C_{t} is defined as a positive number).
For sandstones, this results in , or 0.067 C_{0}. The 15 uniaxial tensional strength σ_{τ}* is slightly different and is defined as the value at which the maximum stress, σ_{1}, equals zero. From Eq. 21, we get
Other basic properties are not so simply derived.
We must produce from the stress relationships (Eqs. 18 or 21) an equation for a failure envelope that permits us to resolve the shear and normal stresses on a failure plane, its orientation, and an approximation of the internal friction, and simply predict regions of instability. The general envelope shapes seen in Figs. 4 and 6 suggest a form like that proposed by Murrell^{[5]} and Bienlawski^{[3]}:
where A, b, and n are material constants. Because the envelope intersects the horizontal axis when the normal stress equals the tensional strength,
When the normal stress is zero, the envelope intersects the vertical axis at the cohesion value C_{u}. From Eq. 25, this requires
Therefore, the general form for an envelope is
To derive the slope, α, at any point, we note that the envelope is only slowly varying over a small stress range and could be locally approximated by a line. If we use a pseudo-cohesion defined by Eq. 14 for the stress condition, σ_{m}, r we can subtract the same from a slightly different stress condition, σ_{m’}, r’. Solving for α gives
The Hoek-Brown stress criteria allow us to redefine the mean, σ_{m}, and differential, r, stresses
By substituting these relations into Eq. 30 for two stresses σ_{3} and σ_{3} + δ σ_{3}, expanding the result and allowing the stress difference, δσ_{3}, to approach zero (what a pain!), we find
As we found previously in Eq. 16, the normal stress is then,
The cohesion is the shear stress value when σ_{n} equals zero. This will occur for σ_{3} somewhere between zero and −C_{t}. In other words, σ_{n} = 0 for
where β is a value around 0.5. We could substitute this term into Eqs. 33 and 34 and solve for β . However, this results in a rather complicated root to a third-order polynomial. Fortunately, by iteration, we can show that β is relatively constant at about 0.62 with little dependence on m. Using this value of β in Eq. 35 and substituting into the previous equations gives us our cohesion. For a sandstone with m = 15, we get
The definition of our curved envelope in Eq. 12 is not strictly compatible with the Hoek-Brown stress relations. However, we can get an estimate of the exponent, n, by using our tensile and cohesion strengths and some reasonable value of σ_{n} such as σ_{n} = C_{0}. From Fig. 2, we can see that τ is approximately 1.1 C_{0} at this point. From Eq. 28, with m equal to 15,
This value falls within the range of 0.65 to 0.75 suggested by Yudhbir et al.^{[6]} Thus, from a presumed simple relation between σ_{1} and σ_{3}, almost all the necessary parameters can be derived.
Nomenclature
C_{0} | = | uniaxial or unconfined compressive strength, GPa or MPa |
C_{t} | = | tensional strength, GPa or MPa |
C_{u} | = | cohesive strength, GPa or MPa |
m | = | Hoek-Brown strength coefficient |
s | = | Hoek-Brown strength coefficient |
r | = | radius of stress "circle," GPa or MPa |
n | = | strength envelope exponent |
σ_{m} | = | mean stress, GPa or MPa |
σ_{n} | = | normal stress, GPa or MPa |
σ_{shear} | = | shear stress components, GPa or MPa |
α | = | aspect ratio |
β | = | strength factor, numeric |
σ_{1} | = | stress in direction 1, GPa or MPa |
σ_{2} | = | stress in direction 2, GPa or MPa |
σ_{3} | = | stress in direction 3, GPa or MPa |
θ | = | wave propagation angle to symmetry axis |
τ | = | shear stress, GPa or MPa |
A | = | strength material constant |
b | = | strength envelope intercept, GPa or MPa |
References
- ↑ Jaeger, J.C. 1969. Elasticity, Fracture, and Flow: With Engineering and Geologic Applications. London: Methuen and Co.
- ↑ ^{2.0} ^{2.1} Scott, T.E. 1989. The effects of porosity on the mechanics of faulting in sandstones. PhD dissertation, University of Texas at Dallas, Dallas, Texas.
- ↑ ^{3.0} ^{3.1} Bienawski, Z.T. 1974. Estimating the strength of rock materials. J. S. Afr. Inst. Min. Metall. 74 (8): 312-320. http://www.saimm.co.za/Journal/v074n08p312.pdf.
- ↑ ^{4.0} ^{4.1} ^{4.2} ^{4.3} Hoek, E. and Brown, E.T. 1982. Underground Excavations in Rock. Amsterdam, The Netherlands: Elsevier Applied Science.
- ↑ Murrell, S.A.F. 1965. The Effect of Triaxial Stress Systems on the Strength of Rocks at Atmospheric Temperatures. Geophys. J. R. Astron. Soc. 10 (3): 231-281. http://dx.doi.org/10.1111/j.1365-246X.1965.tb03155.x.
- ↑ Yudhbir, Lemanza, W., and Prinzl, F. 1983. An empirical failure criterion for rock masses. Presented at the 5th ISRM Congress, Melbourne, Australia, 10-15 April. ISRM-5CONGRESS-1983-042.
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See also
Pore fluid effects on rock mechanics
Subsurface stress and pore pressure