In 1898, Kirsch published equations describing the elastic stresses around a circular hole that are still used today in wellbore pressure breakdown calculations. These equations are standard instruments used in multiple areas of petroleum engineering, however, the original equations were developed strictly for vertical well settings. In today's common directional or horizontal well situations, the equations need adjusted for both deviation from the vertical plane and orientation to the maximum and minimum horizontal in-situ stress anisotropy. This paper provides the mathematical development of these modified breakdown equations. Examples of the implications in varying strike-slip and pore pressure settings are provided. These examples show conditions where it is not unusual for breakdown pressure gradients to exceed 1.0 psi/ft and describes why certain stages in "porpoising" horizontal wells experience extreme breakdown issues during hydraulic fracturing treatments. The paper also discusses how, in most directional situations, the wellbore will almost always fail initially in a longitudinal direction at the borehole wall, after which the far-field stresses will take over and transverse components can be developed. Tortuosity and near wellbore friction pressure can actually add to forcing the initiation of such longitudinal fractures, which can then have cascading effects on other growth parameters such as cluster-to-cluster and stage-to-stage stress shadowing. Special considerations for highly laminated anisotropic formations, where shear failure of the wellbore may precede or preclude tensile failure, are also introduced. Such failure behaviors have significant implications on near wellbore conductivity requirements and can also greatly impact well production and recovery efforts.
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