Content of PetroWiki is intended for personal use only and to supplement, not replace, engineering judgment. SPE disclaims any and all liability for your use of such content. More information

# Elastic wellbore stress concentration

Stress concentration around the wellbore can create breakouts, fractures, or failures. Understanding the stresses on rocks around wellbores is important to well design.

## 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, σθθ, at the wall of a cylindrical wellbore is given by Eq. 1.

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

Here, θ is measured from the azimuth of the maximum horizontal stress, SHmax SHmin is the minimum horizontal stress; Pp is the pore pressure; ΔP is the difference between the wellbore pressure (mud weight) and the pore pressure, and σΔT is the thermal stress induced by cooling of the wellbore by ΔT. If there is no strain in the axial direction, the effective stress acting parallel to the wellbore axis (σzz) is

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

where ν is Poisson’s ratio. At the point of minimum compression around the wellbore (i.e., at θ = 0, parallel to SHmax), Eq. 1 reduces to

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

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

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

The equations for σθθ; and σzz are illustrated in Fig. 1 for a strike-slip/normal faulting stress regime (SHmax ~ Sv > SHmin) at a depth of 5 km, where the pore pressure is hydrostatic and both ΔP and σΔT are assumed to be zero for simplicity. As indicated in Eq. 4 and illustrated in Fig. 1, at the point of maximum compression around the wellbore, the maximum principal 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, 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

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

Substituting this relation into Eq. 3 demonstrates that σθθmin ~ 0, and it is easy for the wellbore to fail in tension, especially if ΔP and ΔT 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.

## Compressive wellbore failure

Stress-induced wellbore breakouts form because of compressive wellbore failure when the compressive strength of the rock is exceeded in the region of maximum compressive stress around a wellbore (Fig. 1). 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. 2[1] 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.

## 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 azimuths 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.

## Detecting wellbore breakouts and tensile fractures

Wellbore breakouts were first identified by Gough and Bell[2] 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. 3,[3] used to distinguish stress-induced wellbore breakouts from drilling-induced features are as follows:

1. When the caliper tool encounters a breakout, the tool should stop rotating in the well, because it should be engaged in the enlargement.
2. The small diameter measured by the caliper must be equal to the bit size.
3. 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.
4. 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. 4). 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.
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 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. 5. 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. 6), whereas in acoustic images, they are most often seen when they are associated with fluid losses (e.g., Fig. 4). 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. 6 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-slip (SHmax > Sv > SHmin).

## Effects of mud weight and temperature on wellbore stress concentration

The equations for stress around a wellbore shown above (Eqs. 1 and 2) include terms that describe thermal effects as well as the influence of the internal wellbore pressure. In the latter case, ΔP = PmudPp; 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 in an increase in σrr and a decrease in σθθ and σzz; this usually inhibits breakout formation, which explains in part why raising mud weight can often solve wellbore instability problems (see Fig. 7). 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

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

There is no effect at the wall of the hole on either σrr or σzz. Raising the temperature of the mud leads to an increase in σθθ, 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.

## Nomenclature

 Em = membrane efficiency, ratio Pp = pore pressure, MPa, psi, lbm/gal SHmin = least horizontal stress, MPa, psi, lbm/gal SHmax = greatest horizontal stress, MPa, psi, lbm/gal So = cohesion, MPa, psi Sv = vertical stress, MPa, psi ΔP = difference between the pressure of fluid in a well and the pore pressure ΔT = temperature difference between the fluid in a well and the adjacent rock α = Biot poroelastic coefficient θ = angle around the wellbore measured from the SHmax direction, degrees σ = Terzaghi effective stress, MPa, psi σrr = effective normal stress acting in the radial direction, MPa, psi σzz = effective normal stress acting on a plane perpendicular to the z direction, MPa, psi σθθ; = the effective hoop stress, MPa, psi Φb = breakout width, degrees ν = Poisson’ s ratio

## References

1. Zoback, M.D., Moos, D., Mastin, L. et al. 1985. Wellbore Breakouts and In Situ Stress. J. Geophys. Res. 90 (B7): 5523-5530. http://dx.doi.org/10.1029/JB090iB07p05523.
2. Gough, D.I. and Bell, J.S. 1982. Stress Orientations from Borehole Wall Fractures with Examples From Colorado, East Texas, and Northern Canada. Can. J. Earth Sci. 19: 1958-1970.
3. Plumb, R.A. and Hickman, S.H. 1985. Stress-induced borehole elongation: A comparison between the four-arm dipmeter and the borehole televiewer in the Auburn geothermal well. J. Geophys. Res. 90 (B7): 5513–5521. http://dx.doi.org/10.1029/JB090iB07p05513.