## Elastic Wellbore Stress Concentration

1.4.1 Stresses Around a Vertical Well. For a vertical well drilled in a homogeneous and isotropic elastic rock in which one principal stress (the overburden stress, Sv) is parallel to the wellbore axis, the effective hoop stress, agg, at the wall of a cylindrical wellbore is given by Eq. 1.8.

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Fig. 1.16—Log-derived unconfined compressive strength for two wells drilled into sands of the Hemlock formation, Cook Inlet, Alaska, along with laboratory test data (dots). In this case, the strength log was derived from the P-wave modulus (pbVp2).7

a00 SHmin + S H max 2(SHmax SH min) cos " p arr = AP

Here, 0 is measured from the azimuth of the maximum horizontal stress, S,

### Hmax ^Hmin

• 1.8) is the minimum horizontal stress; Pp is the pore pressure; AP is the difference between the wellbore pressure (mud weight) and the pore pressure, and aAT is the thermal stress induced by cooling of the wellbore by AT. If there is no strain in the axial direction, the effective stress acting parallel to the wellbore axis (azz) is
• Sv - 2v(S

Hmax H min

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Fig. 1.17—Crossplots of unconfined compressive strength (top) and angle of internal friction (bottom) vs. GR from log inversion. Dots are laboratory test results. While there is considerable scatter, a clear relationship exists between GR and both parameters.

where v is Poisson's ratio. At the point of minimum compression around the wellbore (i.e., at 6 = 0, parallel to SHmax), Eq. 1.8 reduces to

At the point of maximum stress concentration around the wellbore (i.e., at 6 = 90°, parallel to

SHmin);

The equations for a66 and azz are illustrated in Fig. 1.18 for a strike-slip/normal faulting stress

-v • sHmm) at a depth of 5 km, where the pore pressure is hydrostatic and both regime (Sm

AP and aAT are assumed to be zero for simplicity. As indicated in Eq. 1.11 and illustrated in Fig. 1.18, at the point of maximum compression around the wellbore, the maximum principal

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Angle, degrees with respect to the orientation of S^rra*

Angle, degrees with respect to the orientation of S^rra*

1. 1.18—(Top) The upper plot shows the characteristics of the wellbore stress concentration for a vertical well when the vertical stress is a principal stress. Both the circumferential (oee) and axial (azz) stresses are most compressive at the azimuth of the minimum far-field principal stress, leading to the formation of breakouts at that orientation if the stresses exceed the rock's compressive strength, as shown below. Both are most tensile at the azimuth of the greatest horizontal far-field principal stress, possibly leading to tensile failure at the wellbore wall 90° from the orientation of the breakouts (courtesy GeoMechanics Intl. Inc.).
2. 1.18—(Top) The upper plot shows the characteristics of the wellbore stress concentration for a vertical well when the vertical stress is a principal stress. Both the circumferential (oee) and axial (azz) stresses are most compressive at the azimuth of the minimum far-field principal stress, leading to the formation of breakouts at that orientation if the stresses exceed the rock's compressive strength, as shown below. Both are most tensile at the azimuth of the greatest horizontal far-field principal stress, possibly leading to tensile failure at the wellbore wall 90° from the orientation of the breakouts (courtesy GeoMechanics Intl. Inc.).

horizontal stress is amplified appreciably. If the stress concentration is high enough, it can exceed the rock strength, and the rock will fail in compression. Compressive failures that form in the region around the wellbore where the stress concentration is greatest are commonly called stress-induced wellbore breakouts.

For the stress state assumed in Fig. 1.18, the stress concentration is close to zero at the azimuth of the maximum horizontal stress, SHmax. This is because a strike-slip faulting stress state was used for these calculations. It can be straightforwardly shown that in a strike-slip stress state in which the horizontal stress difference is in equilibrium with the strength of vertical strike-slip faults

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Substituting this relation into Eq. 1.10 demonstrates that aeemm ~ 0, and it is easy for the wellbore to fail in tension, especially if AP and AT are greater than zero. Because the horizontal stress difference is smaller in a normal or reverse-faulting stress state than for a strike-slip stress state, tensile failure is less likely in these faulting regimes unless a wellbore is inclined.

To consider the potential for wellbore failure when a wellbore is inclined to the principal stresses, it is necessary to take into account the magnitudes and the orientations of the principal far-field stresses. Once these stress components are determined, in order to know whether a wellbore is likely to fail, the magnitudes of the stresses around the wellbore must be computed and the results considered in the context of a formal failure criterion. Because the equations that describe the stress concentration around a well inclined to the principal stress axes are complicated, they are usually solved using a computer application designed for the purpose.

The wellbore stress concentration decreases as a function of radial distance from the wellbore wall. Thus, the zone of failed rock will only extend to a certain depth away from the well. Once the rock has failed, however, the stresses are re-concentrated around the now broken-out wellbore, and it is possible (depending on the residual strength of the failed rock, which determines whether it can support stress) that additional failure will occur. One important thing to keep in mind is that even if the rock has failed, it may not lead to drilling difficulties.

1.4.2 Compressive Wellbore Failure. Stress-induced wellbore breakouts form because of com-pressive wellbore failure when the compressive strength of the rock is exceeded in the region of maximum compressive stress around a wellbore (Fig. 1.18). If the rock inside the breakout has no residual strength, the failed rock falls into the wellbore and gets washed out of the hole. The shape of these cuttings can be diagnostic of the mode of wellbore failure. Assuming (for the sake of discussion) that a Mohr-Coulomb failure criterion is appropriate for relatively brittle rocks, Fig. 1.198 shows the potential shear failure surfaces for the indicated stress field (left), and the zone of initial failure for a given cohesive strength, So (right). Comparison of the wellbore cross sections with the failure trajectories suggests that the surface of some breakouts is defined by a single shear fracture. It also has been demonstrated that wider and deeper breakouts will form as the maximum horizontal stress increases or as rock strength or mud weight decreases. While there is an increase in the stress concentration at the back of the breakout once it forms, any additional failure caused by that new stress concentration will result in an increase in breakout depth but will not change the width.

In a vertical well, breakouts are centered at the azimuth of minimum horizontal stress SHmin because this is where the compressive hoop stress is greatest. Hence, one can directly deduce the orientation of the in-situ stress tensor from the observation of breakouts. In inclined wells or in vertical wells where one principal stress axis is not parallel to the wellbore, breakout orientations are a function of both the orientations and the magnitudes of the in-situ stresses. Breakouts also may rotate in wells that intersect active shear planes. In both cases, while it is not possible to determine the stress orientation without additional information, it is often possible to determine one or more stress magnitudes.

1.4.3 Tensile Wellbore Failure. It is well known that if a vertical wellbore is pressurized, a hydraulic fracture will form at the azimuth of the maximum horizontal stress SHmax. In some cases, the natural stress state, perhaps aided by drilling-related perturbations such as high mud weight, causes the wellbore wall to fail in tension, generating drilling-induced tensile wall fractures (DITWFs), as previously discussed for a vertical well in a strike-slip faulting environment. These fractures occur only at the wellbore wall (owing to the local stress concentration) and do not propagate any significant distance into the formation. They form 90° from the az-

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1. 1.19—On the left are shown the orientations of potential shear failure surfaces adjacent to a vertical wellbore for SHmax = 45 MPa, SHmin = 30 MPa, AP = 0, and the coefficient of internal friction m = 1.0. On the right is shown the region in which failure is expected for a cohesive strength SQ= 12.5 MPa. The angle 0b is the half-width of wellbore breakout.8 (After M.D. Zoback et al., "Wellbore Breakouts and In-Situ Stress," J. Geophysical Research, Vol. 90, No. B7, 5523; © American Geophysical Union; reproduced/ modified by permission of the American Geophysical Union.)
2. 1.19—On the left are shown the orientations of potential shear failure surfaces adjacent to a vertical wellbore for SHmax = 45 MPa, SHmin = 30 MPa, AP = 0, and the coefficient of internal friction m = 1.0. On the right is shown the region in which failure is expected for a cohesive strength SQ= 12.5 MPa. The angle 0b is the half-width of wellbore breakout.8 (After M.D. Zoback et al., "Wellbore Breakouts and In-Situ Stress," J. Geophysical Research, Vol. 90, No. B7, 5523; © American Geophysical Union; reproduced/ modified by permission of the American Geophysical Union.)

imuths of wellbore breakouts, and in vertical wells they indicate the azimuth of the maximum horizontal stress. As in the case of breakouts, tensile fractures in wells inclined to the principal stresses form at orientations that are a function of the stress magnitude as well as its orientation. In such cases, tensile fractures are inclined with respect to the wellbore axis, thus providing a clear indication that the stresses are not parallel and perpendicular to the well.

1.4.4 Detecting Wellbore Breakouts and Tensile Fractures. Wellbore breakouts were first identified by Gough and Bell9 using 4-arm, magnetically oriented caliper logs acquired with Schlumberger dipmeters. However, to use this information for stress analysis, breakouts must be distinguished from other enlargements such as washouts (in which the entire hole is enlarged) and keyseats (caused by pipe wear or other drilling-related wellbore damage). The criteria, illustrated in Fig. 1.20,10 used to distinguish stress-induced wellbore breakouts from drilling-induced features are as follows. First, when the caliper tool encounters a breakout, the tool should stop rotating in the well, because it should be engaged in the enlargement. Second, the small diameter measured by the caliper must be equal to the bit size. Third, in the case of an inclined well, the direction in which the wellbore is enlarged should not be the same as the direction of wellbore deviation. Finally, neither caliper diameter should be smaller than the bit size, as can occur in zones of keyseats owing to an associated off-centered tool. Failure to utilize criteria such as these can result in interpreting washouts and keyseats as wellbore breakouts.

While breakouts can be detected and used to determine stress orientation in many wells if 4-arm caliper data are carefully analyzed using rigorous criteria, truly unambiguous identification of breakouts requires the interactive analysis of data from full-wellbore scanning tools such as acoustic televiewers, which generate wellbore images that allow a much more detailed investigation of the wellbore wall (Fig. 1.21). These image data have the advantage over caliper data in that it is possible in images to

1. Study detailed variations of breakout orientation with depth.
2. Analyze the precise span of the wellbore's circumference which has failed using wellbore cross sections based on the time of flight of the acoustic pulse.
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1. 1.20—Examples of 4-arm caliper (dipmeter) logs and common interpretations of the borehole geometry. Cal 1-3 and Cal 2-4 indicate borehole diameter as measured between perpendicular dipmeter arms. The shaded regions in the direction of enlargement represent local zones of slightly higher conductivity.10 (After R.A. Plumb and S.H. Hickman, "Stress-Induced Borehole Elongation—A Comparison Between the Four-Arm Dipmeter and the Borehole Televiewer in the Auburn Geothermal Well," J. Geophysical Research, Vol. 90, B6, 5513; © American Geophysical Union; reproduced/modified by permission of the American Geophysical Union.)
2. 1.20—Examples of 4-arm caliper (dipmeter) logs and common interpretations of the borehole geometry. Cal 1-3 and Cal 2-4 indicate borehole diameter as measured between perpendicular dipmeter arms. The shaded regions in the direction of enlargement represent local zones of slightly higher conductivity.10 (After R.A. Plumb and S.H. Hickman, "Stress-Induced Borehole Elongation—A Comparison Between the Four-Arm Dipmeter and the Borehole Televiewer in the Auburn Geothermal Well," J. Geophysical Research, Vol. 90, B6, 5513; © American Geophysical Union; reproduced/modified by permission of the American Geophysical Union.)
3. Unambiguously distinguish stress-induced breakouts from keyseats and washouts. Although electrical image logs can also be used for wellbore failure analysis, it is more difficult to detect and characterize wellbore breakouts in electrical images than in acoustic images.

With the advent of 6-arm, oriented calipers, both those associated with electrical imaging tools and those that are run independently, it is now possible to utilize such data to define the shape of a well and identify oriented enlargements such as those caused by breakouts. To do so, however, these logs must be run in combination with orientation devices. As with 4-arm

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Electrical Image

Fig. 1.21—Example showing that it is possible to identify both tensile cracks and breakouts in acoustical wellbore images (left) and also to identify breakouts in an electrical image (center). Both images show an unwrapped view of the wellbore as a function of depth and azimuth, with azimuths starting at N on the left moving clockwise to E, S, W, and back to N on the right-hand edge. A wellbore cross section of time-offlight data from an acoustic wellbore imaging tool is shown on the right, along with radial lines indicating the azimuthal extent of a wellbore breakout (courtesy GeoMechanics Intl. Inc.).

### Electrical Image

Fig. 1.21—Example showing that it is possible to identify both tensile cracks and breakouts in acoustical wellbore images (left) and also to identify breakouts in an electrical image (center). Both images show an unwrapped view of the wellbore as a function of depth and azimuth, with azimuths starting at N on the left moving clockwise to E, S, W, and back to N on the right-hand edge. A wellbore cross section of time-offlight data from an acoustic wellbore imaging tool is shown on the right, along with radial lines indicating the azimuthal extent of a wellbore breakout (courtesy GeoMechanics Intl. Inc.).

Fig. 1.22—Comparisons of 6-arm caliper data with wellbore cross sections from acoustic televiewer data from within the same interval reveal clearly that the 6-arm tool can be used to detect and orient breakouts.

caliper (dipmeter) data, strict criteria must be defined before using these data to determine stress orientation. An example of a case in which televiewer data was available to validate a 6-arm caliper analysis is shown in Fig. 1.22. Here, the wellbore cross section provided by the acoustic time-of-flight information shows enlargements in the precise orientations of the enlarged parts of the hole detected using 6-arm calipers. It is not possible to constrain breakout widths using 6-arm calipers because the orientation scatter in that data reflects only the variation in position of the centers of the caliper pads where breakouts were detected. Thus, the widths of these two cross sections have little relationship to each other.

Tensile fractures can most easily be seen in electrical image logs (see Fig. 1.23), whereas in acoustic images, they are most often seen when they are associated with fluid losses (e.g.,

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Fig. 1.23—Drilling-induced tensile fractures in Formation Micro-Imager (FMI) data. The fact that the fractures are parallel to the wellbore is an indication that the wellbore axis is parallel to a principal stress direction—in this case, the vertical stress.

### 5,860 -J

1. 1.23—Drilling-induced tensile fractures in Formation Micro-Imager (FMI) data. The fact that the fractures are parallel to the wellbore is an indication that the wellbore axis is parallel to a principal stress direction—in this case, the vertical stress.
2. 1.21). In some very rare cases, wellbores will enlarge in the direction in which tensile fractures are created by excessive amounts of wellbore cooling or extremely high mud weights. This effect has been documented using image logs in cases where stress orientations obtained from caliper logs were interpreted to indicate 90° shifts in stress orientation across bed boundaries. Even if televiewer data are available, enlargements in the direction in which tensile fractures develop can be mistaken for breakouts unless the data are studied carefully.

The cracks seen in Fig. 1.23 occur on both sides of the wellbore at the orientation of the maximum horizontal principal stress in the region, and similar cracks are seen over a ~200-m-long interval of the relatively vertical section of this well. These cracks are principally the result of the natural stress state combined with the additional effects of excess wellbore pressure and cooling, and thus the state of stress implied by the occurrence of these fractures is strike-

1.4.5 Effects of Mud Weight and Temperature on the Wellbore Stress Concentration. The equations for stress around a wellbore shown above (Eqs. 1.8 and 1.9) include terms that describe thermal effects as well as the influence of the internal wellbore pressure. In the latter case, AP = Pmud - Pp; in other words, the mud acts first against the pressure of the pore fluid, and any excess pressure is then applied to the rock. This assumes a reasonably efficient mud cake, and can be modified to account for its absence. If the mud weight is increased, it results

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, arr = AP = 0, possibly leading to large amounts of failure in weak rock. When the mud weight is increased, it increases the radial stress on the wellbore wall and decreases the circumferential stress. This shrinks the Mohr circle without changing its midpoint, leading to a decreased risk of wellbore failure. The increase in effective strength can be as large in weak rock as it is in strong rock, and increases with mud weight at a rate defined by the internal friction (courtesy GeoMechanics Intl. Inc.).

, arr = AP = 0, possibly leading to large amounts of failure in weak rock. When the mud weight is increased, it increases the radial stress on the wellbore wall and decreases the circumferential stress. This shrinks the Mohr circle without changing its midpoint, leading to a decreased risk of wellbore failure. The increase in effective strength can be as large in weak rock as it is in strong rock, and increases with mud weight at a rate defined by the internal friction (courtesy GeoMechanics Intl. Inc.).

in an increase in arr and a decrease in a00 and azz; this usually inhibits breakout formation, which explains in part why raising mud weight can often solve wellbore instability problems (see Fig. 1.24). On the other hand, elevated mud pressures increase the likelihood of drilling-induced tensile wall fractures.

Thermal effects at the wellbore wall in the absence of pore fluid diffusion (that is, for purely conductive heat flow) can be described to first order by ofe = (a E AT)/(1 - v) (1.13)

There is no effect at the wall of the hole on either arr or Raising the temperature of the mud leads to an increase in oee, which enhances the likelihood of breakouts and inhibits tensile fracture formation; on the other hand, cooling the mud inhibits breakouts (at least as long as the mud is kept at a temperature below the temperature of the rock) and increases the likelihood of development of tensile wall fractures. It has recently been noted that leakoff pressure can be increased by wellbore heating, which is consistent with this effect.

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• sakari
What effect does tensile cracks have on wellbore pressure?
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• Codi
How do hydraulic fractures propagate?
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