Analytical solution of the stability of poro-elasto-plastic brittle hollow cylinder subjected to boundary stresses and pore pressures

To obtain an analytical solution for the extension of the failure zone around a wellbore, we consider the problem of a hollow cylinder subjected to boundary stress and pore pressure, and the rock having a brittle elasto-plastic behavior obeying the plastic criterion of Mohr-Coulomb (M-C), which features a sharp transition from intact rock (elastic behavior) to failed rock (plastic). Two cases of effective stress combination have been studied (a) σ'θ > σ'z > σ'r which correspond to a production well and (b) σ'z > σ'θ > σ'r for an injection well. The analytical stress solution in the plastic zone clearly shows that the pore pressure gradient plays a role reducing of rock strength, a kind of "drag force", in case of production well. The condition for unstable plastic extension is demonstrated. Sensitivity studies can be easily conducted using the analytical stress solution. The relationship between the extension of the failure zone and the applied drawdown pressure can be established. The effects of smooth ramp-up or aggressive ramp-up of production on the extension and volume of the failure zone have been studied by varying the outer radius b of the hollow cylinder. The results indicate that a smooth ramp-up can significantly limit the volume of the failure rock for the same drawdown pressure.