Andersonian Faulting Classifications
Fault Regime
S,
Si
Normal
&>
S/^nin
Strikeslip
S^max
SVjmiK
Reverse
Smvtio»
s,
ASH max ASH min 2/3Ap.
where v is Poisson's ratio, and a (= 1  Kdry/Kgrain) is the Biot poroelastic coefficient, which varies between zero for a rock that is as stiff as the minerals of which it is composed and one for most sediments, which are much softer than their mineral components. It is important to note that Eq. 1.1 cannot be used to calculate the relationship between pore pressure and stress in the Earth that develops over geological time because in that case the assumptions used to derive the equation are not valid.
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1.2.4 Effective Stress. The mathematical relationship between stress and pore pressure is defined in terms of effective stress. Implicitly, the effective stress is that portion of the external load of total stress that is carried by the rock itself. The concept was first applied to the behavior of soils subjected to both externally applied stresses and pore pressure acting within the pore volume in a 1924 paper by Terzaghi3 as atJ  Sij — ôtJpp,
where Oj is the effective stress, Pp is the pore pressure, ôj is the Kronecker delta (ôj = 1, if i = j, ôy = 0 otherwise), and Sj represents the total stresses, which are defined without reference to pore pressure. While it is sometimes necessary to use a more exact effective stress law in rock (op = Sj  ôj a Pp, where a is Biot's coefficient and varies between 0 and 1), in most reservoirs it is generally sufficient simply to assume that a = 1. This reduces the effective stress law to its original form (Eq. 1.2). When expanded, the Terzaghi effective stress law becomes and
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 1.6—In a laterally infinite reservoir where L>>h, the relationship between a change in pore pressure and the resulting change in stress is defined in Eq. 1.1 (courtesy GeoMechanics Intl. Inc.).
 1.6—In a laterally infinite reservoir where L>>h, the relationship between a change in pore pressure and the resulting change in stress is defined in Eq. 1.1 (courtesy GeoMechanics Intl. Inc.).
The concept of effective stress is important because it is well known from extensive laboratory experiments (and from theory) that properties such as velocity, porosity, density, resistivity, and strength are all functions of effective stress. Because these properties vary with effective stress, it is therefore possible to determine the effective stress from measurements of physical properties such as velocity or resistivity. This is the basis for most porepressureprediction algorithms. At the same time, effective stress governs the frictional strength of faults and the permeability of fractures.
1.2.5 Constraints on Stress Magnitudes. If rock were infinitely strong and contained no flaws, stresses in the crust could, in theory, achieve any value. However, faults and fractures exist at all scales, and these will slip if the stress difference gets too large. Even intact rock is limited in its ability to sustain stress differences. It is possible to take advantage of these limits when defining a geomechanical model for a field when other data are not available.
Stress Constraints Owing to Frictional Strength. One concept that is very useful in considering stress magnitudes at depth is frictional strength of the crust and the correlative observation that, in many areas of the world, the state of stress in the crust is in equilibrium with its frictional strength. Because the Earth's crust contains widely distributed faults, fractures, and planar discontinuities at many different scales and orientations, stress magnitudes at depth (specifically, the differences in magnitude between the maximum and minimum principal effective stresses) are limited by the frictional strength of these planar discontinuities. This concept is schematically illustrated in Figs. 1.7a and 1.7b. In the upper part of the figure, a series of randomly oriented fractures and faults is shown. Because this is a twodimensional (2D) illustration (for simplicity), it is easiest to consider this sketch as a map view of vertical strikeslip faults. In this case, it is the difference between aHmax (SHmax  Pp) and oHmin (SHmin  Pp) that is
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limited by the frictional strength of these preexisting faults. In other words, as aHmax increases with respect to aHmin, a subset of these preexisting faults (shown in light gray) begins to slip as soon as its frictional strength is exceeded. Once that happens, further stress increases are not possible, and this subset of faults becomes critically stressed (i.e., just on the verge of slipping). The lower part of the figure illustrates using a threedimensional (3D) Mohr diagram, the equivalent 3D case. The frictional strength of faults can be described in terms of the Coulomb criterion, which states that faults will slip if the ratio of shear to effective normal stress exceeds the coefficient of sliding friction (i.e., x/an = p)\ see Fig. 1.8. Because for essentially all rocks (except some shales) 0.6 < ^ < 1.0, it is straightforward to compute limiting values of effective stresses using the frictional strength criterion. This is graphically illustrated using a 3D Mohr diagram as shown in the lower part of Fig. 1.7. 2D Mohr diagrams plot normal stress along the xaxis and shear stress along the >>axis. Any stress state is represented by a half circle that intersects the xaxis at a = a3 and a = aj and has a radius equal to (aj  a3)/2. A 3D Mohr diagram plots three half circles the endpoints of which lie at values equal to the principal stresses and the radii of which are equal to the principal stress differences divided by 2. Planes of any orientation plot within and along the edges of the region between the circles at a position corresponding to the values of the shear and normal stresses resolved on the planes. Planes that contain the a2 plot along the largest circle are first to reach a critical equilibrium.
The critically stressed (light gray) faults in the upper part of the figure correspond to the points (also shown in light gray) in the Mohr diagram, which have ratios of shear to effective normal stress between 0.6 and 1.0. It is clear in the Mohr diagram that for a given value of oHmm, there is a maximum value of oHmax established by the frictional strength of preexisting faults (the Mohr circle cannot extend past the line defined by the maximum frictional strength). The values of S1 and S3 corresponding to the situation illustrated in Fig. 1.7 are defined by v o3 = (S J Pp)/(S3 Pp) = [(«2+1)1/2+ ^ 2 = fb) (1.4) That is, it is the effective normal stress on the fault (the total stress minus the pore pressure) that limits the magnitude of the shear stress. Numerous insitu stress measurements have demonstrated that the crust is in frictional equilibrium in many locations around the world (Fig. 1.9).4 This being the case, if one wished to predict stress differences insitu with Eq. 1.4, one would use Anderson's faulting theory to determine which principal stress (i.e., SHmax, SHmin, or Sv) corresponds to Sj or S3, depending of course on whether it is a normal, strikeslip, or reversefaulting environment, and then utilize appropriate values for Sv and Pp (the situation is more complex in strikeslip areas because Sv corresponds to neither S1 nor S3). Regardless of whether the state of stress in a given sedimentary basin reflects the frictional strength of preexisting faults, the importance of the concept illustrated in Fig. 1.7 is that at any given depth and pore pressure, once we have determined the magnitude of the least principal effective stress using minifracs or leakoff tests (oHmin in a normal or strikeslip faulting case), there is only a finite range of values that are physically possible for oHmax. Eq. 1.4 defines the upper limit of the ratio of effective maximum to effective minimum insitu stress that is possible before triggering slip on a preexisting, welloriented fault. The insitu effective stress ratio can never be larger than this limiting ratio. Therefore, all possible stress states must obey the relationship that the effective stress ratios must lie between 1 and the limit defined by fault slip as shown in Eq. 1.5. 1 < V < V <[(«2+1)1/2+ ^]2 = f(") (1.5) These equations can be used along with the Andersonian definitions of the different faulting regimes (Table 1.1) to derive a stress polygon, as shown in Fig. 1.10. These figures are constructed as plots at a single depth of SHmax vs. SHmin. The shaded region is the range of
Effective Normal Stress, MPa Fig. 1.9—Stress measurements made in brittle rock (dots) reveal that in most of the world, the crust is in a state of frictional equilibrium for fault slip for coefficients of sliding friction between 0.6 and 1.0 as measured in the laboratory (modified after Townend and Zoback4). Effective Normal Stress, MPa Fig. 1.9—Stress measurements made in brittle rock (dots) reveal that in most of the world, the crust is in a state of frictional equilibrium for fault slip for coefficients of sliding friction between 0.6 and 1.0 as measured in the laboratory (modified after Townend and Zoback4). allowable values of these stresses. By the definitions of SHmax and SHmin, the allowable stresses lie above the line for which SHmax = SHmin. Along with the pore pressure, Sv, shown as the black dot on the SHmax = SHmin line, defines the upper limit of SHmax [the horizontal line at the top of the polygon, for which oHmJov = f («)], and the lower limit of SHmin [the vertical line on the lower left of the polygon, for which oJoHmm = f («)]. The third region is constrained by the difference in the horizontal stress magnitudes [i.e., oHmJoHmm < f («)]. The larger the magnitude of Sv, the larger the range of possible stress values; however, as the pore pressure increases, the polygon shrinks, until at the limit when Pp = Sv, all three stresses are equal. It is important to emphasize that the stress limit defined by frictional faulting theory is just that—a limit—and provides a constraint only. The stress state can be anywhere within and along the boundary of the stress polygon. As discussed at length later, the techniques used for quantifying insitu stress magnitudes are not model based, but instead depend on measurements, calculations, and direct observations of wellbore failure in alreadydrilled wells in the region of interest. These techniques have proved to be sufficiently robust that they can be used to make accurate predictions of wellbore failure (and determination of the steps needed to prevent failure) with a reasonable degree of confidence. Stress Constraints Owing to ShearEnhanced Compaction. In weak, young sediments, compaction begins to occur before the stress difference is large enough to reach frictional equilibrium. Therefore, rather than being at the limit constrained by the frictional strength of faults, the stresses will be in equilibrium with the compaction state of the material. Specifically, the porosity and stress state will be in equilibrium and lie along a compactional end cap. The physics of this process is discussed in the section on rock properties of this chapter.
Reverse S^mio Fig. 1.10—This figure shows construction of the polygon that limits the range of allowable stress magnitudes in the Earth's crust at a fixed depth and corresponding magnitude of S„). It is a plot of SHmax vs. SHmin as constrained by the strength of welloriented, preexisting faults. The limits are constrained by Eq. 1.4, with S1 and S3 defined by Andersonian faulting theory, as shown in Table 1.2 (courtesy GeoMechanics Intl. Inc.). Constraints, based on compaction, define another stress polygon similar to the one shown in Fig. 1.10. It is likely that in regions such as the Gulf of Mexico, and in younger sediments worldwide where compaction is the predominant mode of deformation, this is the current insitu condition. Unfortunately, while endcap compaction has been studied in the laboratory for biaxial stress states (a 1 > a 2 ~ a 3), there has been little laboratory work using polyaxial stresses (a1 ^ a2 ^ a3), and there have been relatively few published attempts to make stress predictions using endcap models. Also, it is important to apply endcap analyses only where materials lie along a compaction curve, and not to apply these models to overcompacted or diagenetically modified rocks. If the material lies anywhere inside the region bounded by its porositycontrolled end cap, this constraint can be used only to provide a limit on stress differences. 
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