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.