In anisotropic rocks such as shale, the value of the maximum principal stress required to cause shear failure depends not only on the other two principal stresses, but also on the angle β between the maximum principal stress and the normal to the bedding plane. According to Jaeger's plane of weakness model, for β near 0° or 90°, failure will occur at a stress determined by the failure criterion for the "intact rock", and the failure plane will cut across the bedding planes. At intermediate angles, failure will occur along a bedding plane, at a stress determined by the strength parameters of the bedding plane. Data were analyzed from a set of triaxial (σ2= σ3) compression tests conducted on a suite of shale samples, at different confining stresses, and a range of angles β and it was found that the data could be fit reasonably well with the four-parameter plane of weakness model. Based on these results, a model has been developed for the stability of boreholes drilled in shales. The fully anisotropic Lekhnitskii-Amadei solution is used to compute the stresses around the borehole wall. The Mogi-Coulomb failure criterion is used for the strength of the "intact rock", and the plane of weakness model is used for the strength of the bedding planes. The model can be used to predict the minimum mud weight required to avoid shear failure, for arbitrary borehole orientations and anisotropy ratios. The results show the importance of using a fully anisotropic elastic model for the stresses, and using a true-triaxial failure criterion, in borehole stability analysis.
A fundamental problem in rock mechanics is to predict, based on the stress state, whether or not a rock will "fail". There are several modes of failure, one of the most important being shear failure, in which the initially intact rock breaks along a plane whose orientation is controlled by the orientations and magnitudes of the principal stresses. For isotropic rocks, the simplest and oldest failure criterion is the Coulomb failure criterion (Jaeger et al., 2007), which states that failure will occur if and when
where σ1≥ σ2 ≥ σ3 are the three principal stresses, So is the cohesion, Co is the uniaxial compressive strength, β = 45°+(ϕ/2) is the angle between the normal vector to the failure plane and the maximum principal stress, ϕ = tan−1μ is the angle of internal friction, and μ is the coefficient of internal friction. Many other shear failure criteria have also been proposed (Jaeger et al., 2007; Labuz et al., 2018).
