In performing an elastoplastic analysis for a wellbore or tunnel, one must decide whether to use a 2D or 3D stress analysis and, if using a 3D analysis, whether to use a 2D or 3D strength criterion. This paper presents examples obtained with analytic and numerical solutions by applying these different options to axisymmetric and non-axisymmetric cases. Both linear elastic and elastic - perfectly plastic analyses reveal that accounting for all three stresses can significantly change the results, even when a 2D criterion is applied. Incorporating the intermediate principal stress into the strength criterion (3D) often leads to even greater differences, especially as plastic yielding progresses.
Over the years a number of analytic solutions have been developed to model the creation of a cylindrical hole in rock or soil. The rock or soil behavior is generally described as elastoplastic, to varying degrees of complexity (see the review in Brown, et al. 1983). The goal of these solutions is to predict how the hole will close and how the failed (plastic) zone will expand as the pre-existing support in the hole is reduced. Such solutions can provide guidance for the drilling fluid density required to keep a well stable, or the support required for a tunnel or shaft. The analytic models only apply to cases in which the axis of the hole is parallel to a principal in situ stress, and nearly all the solutions are applicable only for axisymmetric loading of the hole. This is only sufficient for vertical wells in typical sedimentary basins. As horizontal and highly deviated wells have become more common, numerical methods are often employed in order to overcome such geometric limitations. The use of a numerical model also allows one to relax the constraint placed on nearly all the analytic solutions - that only the in-plane stresses (the stresses orthogonal to the hole axis) are considered.
The first analytic solutions for tunnel failure assumed the material would be linear elastic up to its yield strength, described by the Mohr-Coulomb criterion, followed by perfectly plastic behavior (Brown, et al. 1983). Later models pointed out that rock .These are important developments, and comparisons among the results from different models provide useful insights. Unfortunately, these comparisons and insights are only directly applicable to 2D (in-plane stresses only) hole problems, as solutions which incorporate the axial stress are few and specialized (e.g. Florence & Schwer 1978, Risnes et al 1982). This paper instead compares the results of 2D and 3D models, using a relatively simple constitutive law (elastic - perfectly plastic). Both numerical and analytic methods are employed for these comparisons. By examining results for axisymmetric and non-axisymmetric cases, important differences are revealed between the results of 2D vs 3D approaches. There are two ways in which 3D effects can be included in a wellbore or tunnel model. The first is to consider all stress components but still include only the major and minor principal stresses in the strength (yield) criterion.