Near Wellbore Stresses

For a cylindrical hole in  in a thick, homogeneous, isotropic elastic plate subjected to effective minimum and maximum principal stresses ($\sigma _ H$ and $\sigma _h$),
the radial stress,

\[ \sigma_{t}^{'}=\frac{1}{2}\left(\sigma_{H}^{'}+\sigma_{h}^{'}\right)\left(1+\frac{r_{w}^{2}}{r^{2}}\right)-\frac{1}{2}\left(\sigma_{H}^{'}-\sigma_{h}^{'}\right)\left(1+\frac{3r_{w}^{4}}{r^{4}}\right)\cos2\theta-\frac{r_{w}^{2}}{r^{2}}\left(p_{w}-p_{r}\right)
\]


circumferential (hoop) stress,

\[ \sigma_{t}^{'}=\frac{1}{2}\left(\sigma_{H}^{'}+\sigma_{h}^{'}\right)\left(1+\frac{r_{w}^{2}}{r^{2}}\right)-\frac{1}{2}\left(\sigma_{H}^{'}-\sigma_{h}^{'}\right)\left(1+\frac{3r_{w}^{4}}{r^{4}}\right)\cos2\theta-\frac{r_{w}^{2}}{r^{2}}\left(p_{w}-p_{r}\right)
\]


tangential shear stress,

\[ \tau_{r\phi}=\frac{1}{2}\left(\sigma_{H}^{'}-\sigma_{h}^{'}\right)\left(1+\frac{2r_{w}^{2}}{r^{2}}-\frac{3r_{w}^{4}}{r^{4}}\right)
\]


Hoop stress analysis can be used to determine wellbore failures, breakouts and fracture.

Theses stress changes as the location moves away from wellbore.

• Kirsch, G., Die Theorie der Elastizitat und die Beaurforisse der Festigkeitslehre, VDI Z 1857 1968, 42, 707, 1898.
• Jaeger, J. C., Elasticity, Fracture and Flow, 212 pp., Methuen, London, 1961.
• Zoback, M. D., D. Moos, L. Mastin, and R. N. Anderson (1985), Well bore breakouts and in situ stress, J. Geophys. Res., 90(B7), 5523–5530, doi:10.1029/JB090iB07p05523.

Rock Brittleness

Definition:

• Brittle rocks undergo little or no ductile deformation past the yield point (or elastic limit) of the rock.
• Brittle rocks absorb relatively little energy before fracturing.
• Brittle rocks have a strong tendency to fracture.
• Brittle rocks have a higher angle of internal friction

Brittleness in Mining Industry:

Some authors in the mining industry define brittleness index B (loosely defined, but the concept is also called brittleness ratio, brittleness coefficient, or ductility number) as the ratio of uniaxial compressive strength to tensile strength.

\[ B = \frac{\mathrm{compressive}\ \mathrm{strength}}{\mathrm{tensile}\ \mathrm{strength}} = \frac{\sigma_\mathrm{C}}{\sigma_\mathrm{T}}
\]

Altindag (2003) also gives:

\[ B = \frac{\sigma_\mathrm{C} - \sigma_\mathrm{T}}{\sigma_\mathrm{C} + \sigma_\mathrm{T}} \]

Altindag (2002 and 2003) further showed that the most useful measure may be the mean average of compressive and tensile strength:

\[ B = \tfrac12 \times (\sigma_\mathrm{C} + \sigma_\mathrm{T})  \]

Tensile strength is usually correlated with compressive strength, and it may be possible to use just one of these measures as a proxy for brittleness. This is good, because some (most?) labs only measure compressive strength as a standard test, e.g. in routine triaxial rig tests.

Brittleness in Geophysics:

Rickman et al. 2008 proposed using Young's modulus E and Poisson's ratio ν to estimate brittleness. This is appealing to development geophyisicists because elastic moduli are readily available from logs and accessible from seismic data via seismic inversion. Two recent examples are Sharma & Chopra 2012 and Gray et al. 2012. Gray et al. gave the following equations for 'brittleness index' B:

\[ B=50\% \times \left(\frac{E_{\mathrm{min}}-E}{E_{\mathrm{min}}-E_{\mathrm{max}}}+\frac{\nu_{\mathrm{max}}-\nu}{\nu_{\mathrm{max}}-\nu_{\mathrm{min}}}\right) \]

However, this approach remains skeptical, which assumes that a shale's brittleness is (a) a tangible rock property and (b) a simple function of elastic moduli. Computing shale brittleness from elastic properties is not physically meaningful, stated by Lev Vernik stated at the SEG Annual Meeting in 2012.

http://subsurfwiki.org/wiki/Brittleness