A Graphical Analysis-Based Elastoplastic Solution for Undrained Wellbore Drilling Problem in Hoek-Brown Rock Formation

TL;DR: A graphical analysis-based elastoplastic solution for undrained wellbore drilling problem in Hoek-Brown rock formation provides a rigorous and complete solution for evaluating wellbore stability.

Abstract:  An analytical solution is developed for evaluating the stability of an impermeable wellbore in rock formation that obeys Hoek-Brown failure criterion in a hydrostatic stress field. The solution is derived in the framework of the graphical analysis method recently developed by Chen and Abousleiman (2022). The graphical analysis approach can effectively overcome two essential limitations that exist in the implementation of the Hoek-Brown model in the current literature, i.e., first, the mathematical difficulties involved in the treatment of corner singularity of the yield surface; and second, the intermediacy assumption for the axial stress with respect to other stresses. The graph-based solution procedure follows two steps: (1) to geometrically track the stress path trajectory laid out in the deviatoric stress plane; and (2) to formulate the constitutive equations and the radial equilibrium equation. It is found that the effective stress path corresponding to the borehole drilling process is composed of piecewise curved and straight lines in the deviatoric stress plane; it eventually terminates in the major sextant with the Lode angle θ ranging from π/2 to 2π/3 or, on the line of θ = π/2, depending on the mechanical properties of the rock. The analytical solution obtained is a rigorous and complete one for the borehole drilling problem. Wellbore instability is a significant subject in the application of rock mechanics in the oil and gas industry (Santarelli et al, 1986; Yu, 2000). The study and analysis of stresses around wellbores is crucial in the reduction of monetary losses associated with this problem (Yu, 2000; Chen, 2012). Choosing the right model is essential for accurately predicting mud pressure, calculating wellbore stresses, and ensuring safe wellbore drilling (Zoback, 2007; Yu, 2000). Consequently, this can help mitigate two main problems that cause wellbore instability: borehole collapse and borehole fracturing (Santarelli et al, 1986). Estimation of the critical mud pressure is obtained by choosing a failure criterion and using a constitutive model to estimate the stresses induced around the wellbore (Al-Ajmi, 2006).