Semi-inverse method for bending loads in directional wellbores and numerical solution of equivalent stresses

The directional wellbores are subjected to complicated in-situ stresses such as internal pressure, external pressure, axial force, shearing stress, and bending stress(caused by dog leg), which often lead to downhole problems including casing failure, collapse, and bad break. At present, most of the wellbore mechanical models established in related studies are two-dimensional patterns, ignoring the influence of shearing stress, which results in underestimating the equivalent stress of the wellbore. In this paper, a three-dimensional mechanical model for directional wellbores is established, integrating the loads including internal pressure, non-uniform in-situ stress, axial force, shearing stress, and bending stress. On the basis of this model, the calculation formula for Mises equivalent stress of wellbore is derived, and the effects of such factors as non-uniform in-situ stress, shearing stress, rock elastic modulus, Poisson's ratio, internal pressure, and azimuth on the wellbore equivalent stress are analyzed. When two axial in-situ stresses are equal, the results show that the non-uniformity of the third axial in-situ stress increases by 18%, the Mises equivalent stress of the casing increases by 36%, and the equivalent stress increases remarkably with the increase of non-uniformity. When the three axial in-situ stresses are totally different, the Mises equivalent stress of the casing increases by 38% for a 17% increase in non-uniformity. With same non-uniformity, the casing stress is larger, resulting in less safe wellbore. When the shearing stress of 7.3% of the peak normal in-situ stress is applied to the wellbore model, the Mises equivalent stress of the casing, cement sheath and wellbore rock increases by 2.84, 2.79 and 3.75 times, respectively. Mises equivalent stress of the casing is inversely proportional to the elastic modulus and Poisson's ratio of borehole rock. This research can establish the accurate failure criterion for directional wellbore section, thus, preventing or reducing the downhole complications.It can also provide a new idea for solving Mises equivalent stress of directional wellbore section under complicated in-situ stresses.