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, along with examples of the implications in varying strike-slip and pore pressure settings. 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.
In vertical wells it is relatively easy to accept that a tensile fracture will initiate along the axis of the well and propagate in the direction of maximum horizontal stress, opening perpendicular to the minimum stress. As the industry has increasingly moved to horizontal well completion and stimulation, the paradigm that fractures will orient in the plane of the maximum and intermediate stress of the earth tensor, and continue to open against the minimum horizontal stress, has been retained. Fieldscale development plans are based on the expectation of nearly parallel vertical fractures that are orthogonal to the planned borehole (transverse), or set at a predictable angle to the well axis. Azimuth or strike of the fracture is expected to be given by the azimuth of maximum horizontal stress. When the horizontal stress anisotropy is low, a more complex system of possibly orthogonal fractures, or some complex joint network, is expected.