# Kirsch Solution

## Definition - What does Kirsch Solution mean?

Kirsch's solution is derived from the theory of elasticity to calculate the stresses around a circular excavation such as a borehole. The stress levels will define the loading on the borehole wall, and the rock strength required to resist the load. The Kirsch equation is used mostly to model fracture initiation in the oil and gas industry.

The model assumes that the borehole is penetrating, i.e, fluid is pumped into the formation, or non-penetrating, i.e. a mud cake is preventing filtrate losses. The Kirsch solution assumes that multiple factors such as isotropic stress, deviatoric stress, wellbore pressure, and pore pressure are acting independently.

## Trenchlesspedia explains Kirsch Solution

The analysis of stress concentration was studied by Ernst Gustav Kirsch for a linear elastic solution for stresses around the hole in an infinite plate. Kirsch's solution can be used to determine the maximum horizontal stress (sH) which is an important parameter for modeling wellbore stability and drilling optimization.

The stresses are generally compressive, anisotropic and non-homogenous. The magnitude of in-situ maximum horizontal stress is determined using Kirsch equation using parameters such as minimum horizontal stress, unconfined compressive strength, borehole breakouts, friction angle, and cohesion. It can also be determined using breakdown pressure from the formation XLOT test.

XLOT with fracture reopening test was based on Kirsch solution for a circular hole when it is subjected to internal pressure in a medium that is isotropic, homogenous and linearly elastic. It is assumed that the reopening occurs when the minimum tangential stress on the borehole wall is canceled out by fluid pressure on the borehole wall.

Stresses in underground formations are not uniform and change in magnitude based on direction.