Radial stress around vertical wellbore

Input(s)

\(S_{\text {hmax }}\): Maximum Horizontal Stress (psi)

\(S_{\text {hmin }}\): Minimum Horizontal Stress (psi)

\(P_{o}\): Pore Pressure \((\mathrm{psi})\)

\(\mathrm{R}\): Radius of Wellbore (ft)

r: Relative Position to Centre (ft)

\(\theta\): Azimuth of \(\mathrm{S}_{\mathrm{hmax}}(\mathrm{rad})\)

Output(s)

\(\sigma_{r r}\): Stress (psi)

Formula(s)

\(\sigma_{\mathrm{rr}}=0.5 *\left(\mathrm{~S}_{\mathrm{hmax}}+\mathrm{S}_{\mathrm{h} \text { min }}-2 * \mathrm{P}_{\mathrm{o}}\right) *\left(1-\left(\frac{\mathrm{R}^{2}}{\mathrm{r}^{2}}\right)\right)+0.5 *\left(\mathrm{~S}_{\mathrm{h} \max }-\mathrm{S}_{\mathrm{h} \min }\right) *\left(1+\left(\frac{3 * \mathrm{R}^{4}}{\mathrm{r}^{4}}\right)-\left(\frac{4 * \mathrm{R}^{2}}{\mathrm{r}^{2}}\right)\right) * \cos (2 * \theta)+\left(\frac{\mathrm{P}_{\mathrm{o}} * \mathrm{R}^{2}}{\mathrm{r}^{2}}\right)\)

Reference(s)

Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 170.