A lined hole in a viscoelastic rock under biaxial far-field stress

In this paper, we revisit the problem of an infinite, homogeneous, isotropic, linear viscoelastic medium containing a lined circular hole in the general setting of a biaxial far-field stress and for a variety of viscoelastic models for the medium and the lining. The goal is to provide reliable ready-to-use tool to analyze the plane-strain problem of a borehole in viscoelastic rock for any type of viscoelastic model that can be expressed in differential form. The lining is modeled as an isotropic viscoelastic ring. A constant biaxial loading is applied at infinity and/or a time-dependent pressure is prescribed at the inner boundary of the lining. Two distinct scenarios are considered: (i) the hole is excavated before the lining is inserted, i.e. the radius of the hole is larger than the outer radius of the ring, and (ii) the lining is set up directly and a perfect bond is assumed between the lining and the viscoelastic material. These problems are solved using the correspondence principle. The Laplace transformed stresses and displacements in both the lining and the rock are calculated. An inverse Laplace transform is then used to obtain the time domain solution. The operations of space integration and Laplace transform are performed analytically. The obtained new solution in the Laplace domain is compared with published results as well as a numerical solution by the finite element method (for the elastic transformed problem). The Laplace inversion procedure is done either analytically or numerically. For the sake of illustration, examples are given for the cases where the viscoelastic rock responds elastically in dilation and viscoelastically in shear. Five classical viscoelastic models in shear are notably considered. The accuracy of the approach for viscoelastic problems is demonstrated by comparing selected results with a few available analytical solutions for the simpler case of a hydrostatic load at infinity and an incompressible rock. We notably clarify the validity of a number of solutions described in the literature. The efficiency of the solution presented here is illustrated by several numerical examples. For reproducibility, a Mathematica script containing all the derived solutions is provided as supplementary material.