A stress-history dependent analytical solution is presented for analysis of borehole stability in un-consolidated media. The stress distribution around a yielding borehole is calculated, and different permeabilities may be introduced for the plastic (yielded) and elastic zones. We present a sensitivity analysis of important factors, including wellbore pressures, reservoir pore pressure and stress, and material properties (intact and yielded).
Borehole stability is an important issue in the petroleum industry; its effect on drilling budgets is significant [Woodland, 1988]. Various rupture and yield modes may be induced around a borehole under different load states; herein, we propose that plastic shear yield and tensile opening are the dominant mechanisms in weak rocks. We assume that chemical effects are minimal. In this article, we focus on potentially controllable parameters such as wellbore pressure and residual strength, rather than on the uncontrollable initial state parameters.
A strain-weakening material strength model is used to account for the fact that even after yield, some strength remains, even if cohesive strength is almost totally destroyed. This "residual" strength is largely frictional, and can be described by a Mohr-Coulomb criterion of the same type used to describe the "peak" strength. Strain-weakening models provide more realistic results than brittle failure models (conservative), perfectly plastic models (non-conservative), or strain-hardening models (also non-conservative).
Unexpected hydraulic fracture may be responsible for circulation fluid losses in regions of low lateral stress or abnormal pore pressure. Yield can reduce the rocks intact strength, and hydraulic fracture can be induced at a lower wellbore pressure than for intact rock [Wang et. al., 1990). Our analyses show that expected breakdown pressures can be much lower than those predicted by an elastic model. It would seem that the study of realistic models that incorporate yielding could help drilling engineers design mud, drilling and casing programs to reduce the chances of blowout arising from hydraulic fracture, as well as enhance wellbore stability.
Only recently has borehole stability been formulated as a non-linear geomechanical problem and most theories remain unconfirmed by experimental and field data. Models predicting shear rupture by elastic theory have been questioned [Santarelli et. a1., 1986). These authors suggested that a stress-dependent Young's modulus is a more appropriate model. Bell and Dusseault  note that viscoplastic, non-linear elastic, stress-dependent elastic, and elastoplastic strain-weakening models all predict broadly similar stress distributions: a reduction of peak stress and location of peak σ8 within the borehole wall rather than right on the wall surface. From borehole observations, Gough and Bell (1979) proposed that Mohr-Coulomb (MC) shear failure is the major rupture mechanism, although such an interpretation may be questioned when an anisotropic material is encountered [Guenot, 1987] and other rupture mechanisms could lead to similar breakout geometries [Bell and Dusseault, 1990].
Zheng et.al.  proposed model where strenght is scale dependent, in order to avoid the "infinite propagation" of a breakout characteristic of linear elastic models. After initial uniaxial compressive rupture on the borehole wall, a strength increase for the next, unbroken rock element is used, resulting in terminating rupture.