An analytical drained solution based on graphical method for wellbore drilling problem in dilatant Mohr-Coulomb rock formations

Abstract

This paper develops a rigorous analytical solution for the drained wellbore drilling problem in a non-associated Mohr-Coulomb rock formation subjected to hydrostatic initial stress field, on the basis of an efficient graphical method recently proposed in Chen (2024) for the cylindrical cavity expansion problem. Similar to the expansion case, during contraction the deviatoric stress path that is essential for the solution development is found to be composed of a set of piecewise straight lines; it nevertheless always tends to move towards the projection on the deviatoric plane of the negative radial stress axis regardless of the relative magnitude of Poisson's ratio and the friction angle of rock. With the gradual reduction of the internal support pressure during the borehole drilling process, the developed deviatoric stress path will eventually end in a major sextant with Lode angle $\theta$ in between $\frac{2\pi }{3}$ and $\frac{5\pi }{6}$ or on the specific line of $\theta =\frac{5\pi }{6}$ (i.e., the edge of Mohr-Coulomb yield surface). The proposed graphical analysis-based analytical approach effectively removes the stringent yet unnecessary intermediacy assumption for the vertical stress that is commonly adopted in the existing wellbore drilling analysis. Furthermore, it circumvents the use of the traditional cumbersome zoning method that demands the separation of the Mohr-Coulomb plastic zones around the wellbore into multiple distinct ones according to the ordering of the radial, tangential, and vertical principal stresses. Some typical numerical results are presented for the desired wellbore drilling curves and the impacts of the key rock mechanical parameters on the calculated curves are also investigated. The analytical solution obtained by utilizing the present graphical approach is a rigorous and complete one of its kind for the borehole drilling problem involving the classic Mohr-Coulomb constitutive model. It may contribute to better and more accurate prediction and design of mud weight required to maintain the wellbore stability, when compared with the conventional methods for wellbore stability analysis used in the current drilling and operation practice.