Graphical Analysis–Based Elastoplastic Solution for Wellbore Stability Problem in Hoek–Brown Rock under Moderate Nonhydrostatic Far-Field Stress and Undrained Conditions

A semi-analytical solution is proposed 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 is derived within the framework of a recently developed graphical analysis method for the cylindrical cavity expansion problem. The graphical approach can effectively overcome three essential limitations that exist in the implementation of Hoek–Brown model for cavity analyses in the current literature: first, the exclusion of the corner loading case due to the mathematical difficulties of treating the corner singularity related to the nonsmooth potential surface; second, the assumption that the wellbore is subjected to a hydrostatic in situ stress field; and third, the small strain assumption. The graph-based solution procedure presented here contains three steps: (1) geometrically tracking the stress path trajectory laid out in the deviatoric stress plane; (2) formulating the constitutive equations and the radial equilibrium equation in Lagrangian form to calculate the cavity/wellbore responses; and (3) generating the wellbore drilling curves. It is found that, for a moderate value of the earth pressure coefficient, the effective stress path corresponding to the borehole drilling process starts as a straight line in the elastic deformation phase, and then bends and takes a curved path during the plastic phase in the deviatoric stress plane. It eventually terminates either in the major sextant with the Lode angle θ ranging from π/2 to 2π/3, or on the specific line of θ=π/2, when the wellbore pressure is fully released (equal to zero), depending on the mechanical properties of the rock. The developed graphical solution has been applied to determine the critical mud pressure required for maintaining the wellbore stability during drilling.