Modelling Of Damaged Zones Around Boreholes Using A Radius Dependent Young's Modulus
Authors: Nawrocki P.A. , Dusseault M.B. 
Published: March 2010

Abstract
A new "damage mechanics" type model to calculate stresses around circular boreholes in plane strain is presented. The model is based on the assumption that Young's modulus variation with radial distance can be simply described, i.e., E=f(r). Such variations in stiffness arise naturally during and after drilling from damage such as cohesion loss, frictional slip of grains and microfissuring, as well as from the stress redistributions that occur. This approach allows convenient and simple hollow cylinder stress calculations for a variety of E=f((σ3) functions for which a formal pressure-dependent modulus model requires iterative integral evaluations [Equation (8)]. Two functions are suggested for E=j(r) dependency: an exponential law and a power law. The stress distributions from these models are compared to other models previously suggested for borehole stresses, and appear reasonable.

Introduction
Either Linear Elastic (LE)(l,2) or Elasto-Plastic (EP)(3,5) rock behaviour models are commonly used in modelling openings in geomaterials. Recently, Santarelli(6,7) added to these by noting that in granular rocks such as sandstones, Young's modulus, E, generally varies with the radial or minor principal stress, σ3 ' resulting in his pressure-dependent modulus model (PDM model). He obtained σ3 distributions which correspond well to experimental results from thick-walled hollow cylinders(4,8); σ3 values near the borehole wall were much lower than those predicted by LE models. However, for cases where the Young's modulus depends explicitly on the confining stress σ3' that is E=E((σ3), a closed-form PDM model solution can be found only if its inverse l/E((σ3) can be integrated, and only Kulhawy's(9) function seems to give a closed-form solution. For materials with a complex E(σ3) function, a numerical stress-strain solution is necessary, but this can be simplified considerably by using a radius-dependent modulus model (RDM model). We develop and test such a non-linear (NL) model in this article, introducing a stiffness which is a function of the radius.
In our model, the mechanical behaviour of material is assumed to be related to the damage that it may have experienced. Damage around a borehole may be expected during drilling, cementation, production of fluids, high pressure injection (perhaps leading to hydraulic fracturing), or thermal effects(10). This damage can be of different types (grain damage, contact slip, cohesion loss, microfissuring, joint slip), and can be expected to result in a reduced Young's modulus; this reduced modulus in turn will result in a redistribution of stresses around the borehole.
In tunnelling, a zone of significantly reduced Young's modulus is generated for at least several radii away from the wall in cases where drill and blast techniques are used, but at some distance the damage effect vanishes and the original modulus obtains. Similarly, a borehole in shale or sandstone is likely surrounded by a limited extent zone of damage arising from the high deviatoric stresses and reduced radial stress caused by borehole creation.