geomechanics · wellbore stability
Hoop Stress (Kirsch, at a Known Pw)
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Inputs
psi
psi
psi
psi
Description
Computes the total and effective tangential (hoop) stress at the wellbore wall, at the azimuth of maximum stress concentration (θ=90° from the SHmax direction, aligned with Shmin), for a given known or planned wellbore pressure. This is the same Kirsch (1898) stress-concentration term Critical Mud Weight (Wellbore Breakout) solves internally when finding the threshold Pw — this calc exposes it directly for an arbitrary/known Pw instead, letting you check the actual stress state at a planned mud weight rather than only the failure threshold.
Variables
| Symbol | Unit | Description |
|---|---|---|
| σθθ | psi | Hoop Stress (Total) |
| σθθ' | psi | Hoop Stress (Effective) |
| SH | psi | Maximum horizontal stress — typically the upper bound from the SHmax (Stress Polygon) calc, or an independently known/measured value. |
| Sh | psi | Minimum horizontal stress — typically the Shmin output of the Fracture Gradient (Eaton 1969) calc. |
| Pw | psi | Actual or planned wellbore pressure (mud weight expressed as pressure) at the depth of interest — unlike Critical Mud Weight, which solves for the threshold Pw, this is a known/assumed value you supply directly. |
| Pp | psi | Pore pressure at the depth of interest. |
Assumptions
- Vertical wellbore, with the wellbore axis aligned with Sv (one of the three principal stress directions) — no stress-tensor rotation is needed
- Linear elastic rock behavior (Kirsch's solution assumes elasticity)
- Evaluated at θ=90° only (the maximum-hoop-stress azimuth) — the hoop stress at other azimuths, including the minimum at θ=0°, is not computed here
Limitations
- Vertical-well case only — a deviated or horizontal wellbore requires resolving the full 3D stress tensor into the borehole's local coordinate system (Peska & Zoback 1995), which is not included here
- Reports stress at a single azimuth (θ=90°) — does not determine the angular extent (breakout width) over which the rock actually fails, which requires evaluating the failure criterion at other azimuths as well
- Does not itself apply a failure criterion — pair the effective hoop stress output against Mohr-Coulomb Failure Criterion's critical stress/UCS, or Critical Mud Weight's threshold, to assess actual breakout risk
Use Cases
- → Stress-state check at a planned mud weight: Evaluate the actual hoop stress at a specific planned or historical Pw, rather than only the failure-threshold Pw that Critical Mud Weight solves for.
- → Margin-to-failure screening: Compare the computed effective hoop stress directly against Mohr-Coulomb Failure Criterion's critical stress or UCS output to quantify how close a given mud weight is to breakout, not just whether it exceeds the threshold.
- → Diagnosing observed breakout at a known historical Pw: Given the mud weight actually used when breakout was observed on image logs, compute the hoop stress that was acting at the time to back-check assumed stress or strength inputs.
Related Calculations
Region Notes
Permian Basin
Wolfcamp shale intervals with significant stress anisotropy commonly show a meaningful gap between planned mud weight and the critical threshold — this calc quantifies that margin directly rather than only flagging pass/fail.
Global
Useful anywhere Critical Mud Weight's threshold Pw has already been computed and the question shifts to 'how much margin does my actual planned mud weight have', rather than 'what is the minimum mud weight'.
References
Primary source
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