The state of stress in underground formations is a function of pore pressure, overburden stress, and tectonic environment. When a hole is drilled through a rock formation, the stressed rock is replaced with a drilling mud. Estimating drilling induced stresses is central to the determination of optimal drilling mud weight. Analytical and numerical solutions to determine the stress field around boreholes in elastic rocks currently exist. However, there exists a lack of reliable numerical methods able to accommodate the highly complex situations present in compactible rocks.
This paper advances a general framework for estimating stresses around boreholes drilled in weak and compactible rocks. A review of the germane forces followed by development of a three dimensional Finite Element model is presented. Rock deformation is modeled using the Extended-Sandler-Rubin (ESR) cap model, a sophisticated material model which includes a non-linear shear failure surface and a second yield surface (cap) to account for inelastic compaction at stress state lower than those required to induce shear failure.
This model couples fluid flow with rock deformation, a major improvement over existing analytical solutions. As such, it can estimate borehole stresses for cases with constant and varying pore pressure. Furthermore, injection and production induced stresses can also be found using this model. Comparisons between the proposed model and the existing analytical elastic solutions in the literature for constant and varying pore pressure are presented.
During drilling operations, load bearing formation rock is replaced with drilling mud. This leads to stress redistribution in the vicinity of the well. Non-existent prior to drilling, deviatoric stress forms around the well. Comparatively, weak rock formations have less resistance to deviatoric stress. Hence, determination of the new stress distribution is central to preventing mechanical wellbore stability problems.
Linear Elastic Solutions
Kirsch  gave the first general solution to find stresses in a hollow linearly elastic cylinder subjected to internal and external radial stresses. Since then, these solutions have grown tremendously. Nowadays, there is a linear elastic solution for almost every situation one might encounter in the field, including with/without fluid flow, poroelastic effects, and so on .