Stresses Around a Wellbore Due to Internal Pressure and Unequal Principal Geostatic Stresses
Authors: Galle E.M. , Wilhoit J.C. 
Published: December 2007

Abstract
A three-dimensional photoelastic study was made to determine the stress state around the wall and bottom of a wellbore due to fluid pressure within the wellbore and unequal principal geostatic stresses. Models simulating the wellbore were made from a phthalic anhydride-cured epoxy resin. The frozen stress technique of three-dimensional photoelasticity was used to determine the stresses within the model. Two separate problems were solved:hydrostatic pressure on all sides of the model with the exception of the inside of the hole, anduniaxial compression perpendicular to the center line of the hole.
The results are presented in the form of contour curves for each stress component. The curves cover a region which extends several hole radii from the bottom and wall of the hole. By proper use of these results, the state of stress may be calculated for any point on the wall or bottom of the hole, as well as for any point within the material that surrounds the hole. A systematic method is given for calculating these stress components for any combination of fluid pressure within the wellbore and system of unequal principal geostatic stresses, provided one of the principal geostatic stresses is parallel with the wellbore. The results show that stresses around the wall and bottom of a wellbore induced by an unequal system of principal geostatic stresses, are appreciably different from those induced by geostatic stresses that are hydrostatic. For unequal principal geostatic stresses, the experimentally determined stresses on the wall of the hole several radii from the bottom are in good agreement with the stresses calculated by elastic theory.

Introduction
Interest in the problem of determining the stresses around and near the bottom of a wellbore, due to geostatic loading and fluid pressure within the wellbore, originated in the petroleum industry. The stresses on the wall of the wellbore are of interest due to their relation to lost circulation during drilling, the fracturing of formations during squeeze cementing and hydraulic fracturing of producing formations. The stresses on the bottom of the hole are of interest to the drilling segment of the industry because to produce a hole in the earth it is necessary to remove material from the bottom of the hole. This removal of material is produced by structural failure of the rock, whatever the criterion for failure may be. Cunningham and Eenink have shown that for certain conditions the effect on drilling rate of the bottom-hole stresses caused by a hydrostatic overburden pressure is small compared to other effects. This question, however, needs further experimental and analytical investigation. Much information concerning the stresses on the walls of wellbores has been published considering both equal and unequal principal geostatic stresses. The problem of determining the stresses around the bottom of the wellbore is considerably more complicated. However, previous investigators have obtained numerical and experimental solutions to the problem for equal principal geostatic stresses. Whitworth and Woods have obtained numerical solutions. Word obtained a three-dimensional photoelastic solution, Durelli and Deily obtained surface stresses only by photoelastic means, and Cheatham and Wilhoit determined the stresses around the bottom of a short cylindrical cavity. While Miles and Topping, McGuire, Harrison and Kieschnick, and Hubbert and Willis obtained the stresses on the wall of a wellbore for unequal principal geostatic stresses, they did not concern themselves with the stresses near the bottom of the hole. The estimates made by Hubbert and Willis of the differences between the principal geostatic stresses indicate that, to obtain a satisfactory solution for the stresses around the bottom of a wellbore, unequal principal geostatic stresses should be considered. This work is directed toward obtaining such a solution. The materials in the earth's crust are nonhomogeneous, permeable and anisotropic.
SPEJ
P. 145^