Closed-form approximations to borehole stresses for weak transverse isotropic elastic media

Abstract

We have derived new closed-form approximations to borehole stresses for weak tilted-transverse-isotropic media with boreholes oriented in the plane of isotropy but not necessarily along a principal stress direction. By introducing Green's three anisotropic parameters, ω1,ω2 and ω3, into Lekhnitskii's formalism for anisotropic elastic media to calculate stresses around boreholes subjected to internal pressure and in situ stress, we first have recasted the general borehole stresses into more compact expressions that are useful to highlight explicitly the impact of the elastic anisotropy. We have shown the equivalence between Lekhnitskii's and Green's equations for the in-plane components and extended Green's expressions to the anti-plane problem. By linearizing the expressions for weak anisotropy degree, i.e. ∣ωi∣≪1, we have shown that the borehole stress expressions are the sum of the isotropic “Kirsch” solution and additional terms that depend on the anisotropic parameters ω1+ω2 and ω3. The additional terms have azimuthal/radial functional dependence of higher orders (cos4θ,sin4θ) and (1/r2,1/r4,1/r6) for in-plane components, and, (cos3θ,sin3θ) and (1/r2,1/r4) for anti-plane components. Our verification examples show that the new borehole stress expressions provide good approximations not only for weak anisotropy but also moderate and strong anisotropy in certain cases.