A New Model For Predicting Time-dependent Failure of Shales: Theory And Application
Authors: Hawkes C. , McLellan P. 
Published: March 2010

Abstract
The effects of pore pressure penetration and time-dependent rock strength on the growth of the yielded zone around a borehole are analysed using an elastic-brittle-plastic model. Solutions for two possible cases are developed:no change in permeability, anda significant increase in. permeability upon yielding.
The extent of the yielded zone is sensitive to a number of mechanical parameters, of which the residual strength of the rock is most critical. In bothcases, the rate of yielded zone growth depends strongly on formation permeability (both prior to and after yielding) and mud filtrate viscosity. In case (2), the initial extent of the yielded zone is also critical to the rate at which it will grow. Model predictions are compared to field data from a western Canadian setting where time-dependent borehole enlargement occurred in a shale interval.

Introduction
Borehole instability in shales is a frequent problem experienced during drilling. Although the potential destabilizing effects of clay hydration and swelling have long been known and accepted, it is now recognized that borehole instability in shale sections is also strongly dependent on mechanical parameters, i.e., the state of effective stress around the borehole and the strength of the shale. The following discussion describes the mechanical factors affecting borehole stability. Hydration effects are only considered indirectly, as they may affect the strength of the rock.

Elastic-brittle-plastic Borehole Stability Model
A number of non-linear borehole stability models have previously been developed. These include damage mechanics models(1, 2), bifurcation models(3), non-continuum models(4, 5), linear elastic borehole breakout models(6), strain-hardening elastoplastic models(7) and strain-weakening, elastic-brittle-plastic models(8–10). The paper presented here is a new development of the latter type of model.
The defining feature of the elastic-brittle-plastic model is the assumption that, once a rock's peak compressive strength has been exceeded when stresses around the borehole increase during excavation, the rock yields but does not fall into the borehole. Instead, an annulus of weakened rock (referred to as the yielded or plastic zone) develops around the well, peak stress is redistributed away from the borehole surface, and eventually a stable configuration is achieved (Figure 1). Since the yielded zone will be susceptible to spalling due to pressure surges during trips and mechanical erosion by the drillstring, the larger this zone is the greater the likelihood that hole enlargement and instability-related problems will occur.
In order to develop a closed-form solution to this problem, it is necessary to make a number of assumptions concerning the mechanical behaviour of the rock. Firstly, it is assumed that deformation is linear elastic until peak stress is reached, then strain weakening occurs instantaneously and stress is reduced to a residual level (Figure 2). It is also assumed that peak and residual strength of the rock can be represented by linear Mohr-Coulomb criteria, plane strain conditions prevail, the material is homogeneous and isotropic, boundary stresses are uniform (so that the problem is axisymmetric), and the borehole axis is parallel to one of the principal in situ stresses.