Kirsch solution components

The complete Kirsch solution assumes independent action of multiple factors: far-field isotropic stress, deviatoric stress, wellbore pressure and pore pressure.

1) The solution for a compressive isotropic far field stress $\sigma _{\infty }$ is shown in Fig. 6.4. The presence of the wellbore amplifies compressive stresses 2 times $\sigma_{\theta \theta}/\sigma_{\infty}=2$ all around the wellbore wall. The presence of the wellbore cavity also creates infinitely large stress anisotropy at the wellbore wall $\sigma_{\theta \theta}/\sigma_{rr}= \infty$ all around the wellbore wall. Stresses decrease inversely proportional to $r^2$ and are neglible at $\sim$4 radii from the wellbore wall.

Figure 6.4: Kirsch solution for far field stress $\sigma _{\infty }$.

2) The solution for fluid wall pressure in the wellbore $P_W$ is shown in Fig. 6.5. We assume a non-porous solid now. This assumption is relaxed later. Wellbore pressure adds compression on the wellbore wall $\sigma_{rr} = +P_W$, and induces cavity expansion and tensile hoop stresses $\sigma_{rr} = - P_W$ all around the wellbore.

Figure 6.5: Kirsch solution for wellbore pressure $P_W$.

3) The solution for a deviatoric stress $\Delta \sigma$ aligned with $\theta = 0$ is shown in Fig. 6.6. The deviatoric stress results in compression on the wellbore wall $\sigma_{\theta \theta} = 3 \Delta \sigma$ at $\theta = \pi/2$ and $3\pi/2$, and in tension $\sigma_{\theta \theta} = - \Delta \sigma$ at $\theta = 0$ and $\pi$. The presence of the wellbore amplifies compressive stresses 3 times $\sigma_{\theta \theta}/\sigma_{\infty}=3$ at $\theta = \pi/2$ and $3\pi/2$. The variation of stresses around the wellbore depend on harmonic functions $\sin$ and $\cos$ of $\theta$.

Figure 6.6: Kirsch solution for far-field deviatoric stress $\Delta \sigma$.

4) The last step consists in assuming a perfect mudcake, so that, the effective stress wall support (as shown in Fig. 6.5) is $(P_W - P_p)$ instead of $P_W$.