A Complete Rigorous Semianalytical Graphical Solution for the Undrained Wellbore Stability Problem in Elastoplastic Hoek–Brown Rock under Nonhydrostatic In Situ Stresses

This work develops a rigorous semianalytical solution for evaluating the stability of an impermeable wellbore in a rock formation that obeys the Hoek–Brown failure criterion in a nonhydrostatic stress field. The solution expands a previous contribution by the authors by overcoming the intermediacy assumption limitation for the vertical stress. It hence provides a complete solution for the cavity contraction problem for associated Hoek–Brown rock. The developed solution has the practical importance of effectively predicting the critical mud pressure that is essential to maintain the stability of wellbores. This solution is obtained following the recently proposed graphical method, which has the advantage of overcoming the mathematical difficulties caused by the singularity problem encountered in plasticity models with angular yield/potential surfaces such as Mohr–Coulomb and Hoek–Brown, and freeing the axial stress from the intermediacy assumption. With the use of this method, a generalized solution for arbitrary values of K0 is obtained, first by tracking the stress path as it goes through different states of stress until it reaches the stress state corresponding to the reduced wellbore support pressure and then deriving the necessary constitutive equations for the pertaining sectors in the deviatoric plane. The equations developed herein are subsequently used to obtain the wellbore drilling curves for various Hoek–Brown parameters as well as for different values of K0. The results indeed demonstrate the capacity of this novel method to overcome the previous limitations in the literature, by showing how the intermediacy assumption for the vertical stress cannot be applied universally to all K0 scenarios.