The Mechanical Behavior of Uncased Wellbores Situated in Elastic/Plastic Media Under Hydrostatic Stress
Authors: Gnirk Paul F. 
Published: December 2007

Abstract
Analytical consideration is given to the elastic/perfectly plastic behavior of uncased wellbores that are situated in rock formations subjected to hydrostatic states of field stress. Plane strain is assumed, and the rock is supposed to be Plane strain is assumed, and the rock is supposed to be incompressible and to obey the linear Mohr-Coulomb criterion of plastic yield. Using elasticity and plasticity theories, equations are developed to plasticity theories, equations are developed to permit calculation of the wellbore pressure required permit calculation of the wellbore pressure required to prevent plastic yielding of an uncased wellbore wall and of the magnitude of the wall deformation if plastic yielding does occur. The state of induced stress corresponding to the development of a plastic annulus of circular shape around the wellbore is obtained. From analytical results, plastic instability and collapse of a wellbore are discussed. In general, the calculations for the pressures required to prevent plastic yielding and potential instability of wellbores plastic yielding and potential instability of wellbores are quantitatively reasonable.

Introduction
The mechanical behavior of an uncased wellbore situated at depth in a highly deformable rock medium is an important topic. In the usual drilling situation, the mud column exerts a fluid pressure on the wall of the wellbore, while the undisturbed rock formation is subjected to an interstitial-fluid pore pressure and a natural state of field stress. As a consequence of the fluid pressures and the natural field stress, the state of induced stress in the rock formation immediately adjacent to the wellbore may be of sufficient magnitude to cause plastic yielding of rock. If deformation results in a noticeable contraction of the wellbore, the motion of the drill bit may be restricted, or the emplacement of the casing after drilling ceases may be hampered. If the deformation results in a large expansion of the wellbore, the rock formation may fracture and result in lost circulation. Ideally, knowing the range of drilling mud weights that could be utilized for particular depth-rock formation combinations particular depth-rock formation combinations without causing plastic deformation of the wellbore wall would represent an important contribution to drilling technology.
The deformational behavior of underground openings has been of considerable interest to the mining industry. Fenner, Labasse and Morrison and Coates have obtained analytical solutions for the states of stress around vertical- and horizontal-cylindrical openings situated in elastic/perfectly plastic, cohensionless media that obey the Coulomb failure criterion. The analyses are generally extended to consider the influence of concrete or steel liners in the openings. Pasley and Coates have considered analytically the state of stress induced by the flow or interstitial pore fluid into a wellbore for the case in which pore fluid into a wellbore for the case in which the field stress is uniform and the rock obeys the Mohr-Coulomb failure criterion. The rock strength required to prevent the initiation of plastic failure is related to the formation pore pressure and the extent of plugging around the wellbore.
The wellbore deformation problem is very complex in that the deformations are usually nonlinear and time-dependent, and the virgin field stresses are nonuniform. We considered the deformations to be large, but not time-dependent, and assumed the natural field stress to be of a hydrostatic nature. The rock is taken to be isotropic and homogeneous and to obey the linear Mohr-Coulomb criterion at yield. In the rock formation adjacent to the wellbore, the deformation parallel to the wellbore axis is assumed to be negligible as compared to the radial deformation of the wellbore wall. Under these assumptions, it is possible to determine the range of mud weights required to prevent the initiation of plastic yielding at the wellbore wall and to obtain a reasonable estimate of the magnitude of the wall deformation should yielding occur, as well as the extent of the plastic region around the wellbore. By utilizing the results of these considerations, the potential instability of a wellbore, involving collapse, can be discussed with some degree of competence.