Engineering lesson
Minimum and maximum hoop stress around a wellbore
Hoop stress varies around a wellbore because far-field stresses are redistributed by the cavity and boundary pressure. Minimum and maximum values require a consistent effective-stress and angular convention.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Identify the idealized well geometry
The familiar Kirsch-style illustration assumes a circular vertical well with specified far-field horizontal stresses. Its extrema should not be transferred unchanged to a deviated borehole.
The linked minimum-hoop calculator includes its documented pore-pressure, wellbore differential-pressure and thermal terms. Each term has a specified sign.
Keep pressure effects from being counted twice
Wellbore pressure and pore pressure can appear through a differential term and separate effective-stress terms. Replacing one with the other without rederiving the balance can subtract pore pressure twice.
The explorer varies the supplied wellbore-over-pore differential while holding the other stress inputs fixed, using the calculator's exact convention.
Stress is not strength
A minimum hoop stress value alone does not determine a fracture limit. Tensile strength, thermal effects, anisotropy and the relevant failure condition require separate consideration.
Similarly, a high compressive hoop stress needs comparison with an appropriate compressive failure model rather than an arbitrary stress ceiling.
Explore the example
Replayed with the same equation record and calculation engine as /calculators/minimum-hoop-stress-at-the-wellbore-wall. Original synthetic inputs; engine outputs are saved in the downloadable record.
Select one of 5 recorded scenarios. Intermediate states are not interpolated.
Difference Between Wellbore Pressure and Pore Pressure (psi) = 500. sigma_theta_min (psi): 2200.
- sigma_theta_min (psi)
Worked example
Minimum Hoop Stress at the Wellbore Wall
Fixed controls: S_hmin = 6000 psi; S_hmax = 8000 psi; P_o = 3500 psi; S_dt = 300 psi. dP is varied as shown.
Relation and unit basis
sigma_theta_min = 3 * S_hmin - S_hmax - 2 * P_o - dP - S_dt
S_hmin: Minimum Principal Stress in Reservoir (psi). S_hmax: Maximum Principal Stress in Reservoir (psi). P_o: Pore Pressure (psi). dP: Difference Between Wellbore Pressure and Pore Pressure (psi). S_dt: Thermally Induced Stress (psi).
1. Define the baseline
Difference Between Wellbore Pressure and Pore Pressure (psi) = 500
Fixed controls: S_hmin = 6000 psi; S_hmax = 8000 psi; P_o = 3500 psi; S_dt = 300 psi. dP is varied as shown.
2. Select the relation
sigma_theta_min = 3 * S_hmin - S_hmax - 2 * P_o - dP - S_dt
S_hmin: Minimum Principal Stress in Reservoir (psi). S_hmax: Maximum Principal Stress in Reservoir (psi). P_o: Pore Pressure (psi). dP: Difference Between Wellbore Pressure and Pore Pressure (psi). S_dt: Thermally Induced Stress (psi).
3. Substitute the values
S_hmin = 6000; S_hmax = 8000; P_o = 3500; dP = 500; S_dt = 300 → sigma_theta_min = 2200 psi
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
sigma_theta_min (psi) = 2200
This is the published minimum-hoop convention; a strength comparison and orientation model remain separate.
Result and interpretation
sigma_theta_min (psi) = 2200
This is the published minimum-hoop convention; a strength comparison and orientation model remain separate.
The differential-pressure input changes under the published minimum-hoop expression; other terms remain fixed.
Exact scenario data
Replayed with the same equation record and calculation engine as /calculators/minimum-hoop-stress-at-the-wellbore-wall. Original synthetic inputs; engine outputs are saved in the downloadable record.
| Difference Between Wellbore Pressure and Pore Pressure (psi) | sigma_theta_min (psi) |
|---|---|
| 0 | 2,700 |
| 250 | 2,450 |
| 500 | 2,200 |
| 750 | 1,950 |
| 1,000 | 1,700 |
Scroll the table to see all columns. The highlighted row is the worked baseline. Downloads retain full numerical precision.
Test your understanding
Work through the question before revealing its answer. Use the stated assumptions and include units in every numerical step.
1. Why can two hoop-stress formulas differ by pore-pressure terms?
Show the worked answer
They may use different total/effective stress and wellbore-pressure definitions. Reconcile those conventions before comparing numbers.
2. Reproduce the baseline sigma_theta_min (psi) without rounding intermediate values.
Show the worked answer
S_hmin = 6000; S_hmax = 8000; P_o = 3500; dP = 500; S_dt = 300 → sigma_theta_min = 2200 psi. sigma_theta_min (psi) = 2200. This is the published minimum-hoop convention; a strength comparison and orientation model remain separate.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Difference Between Wellbore Pressure and Pore Pressure (psi) changes from 0 to 1000. Other controls remain as stated in the premise. sigma_theta_min (psi) changes from 2700 to 1700.
4. What would be needed to apply this example to a different physical system?
Show the worked answer
Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. S_hmin: Minimum Principal Stress in Reservoir (psi). S_hmax: Maximum Principal Stress in Reservoir (psi). P_o: Pore Pressure (psi). dP: Difference Between Wellbore Pressure and Pore Pressure (psi). S_dt: Thermally Induced Stress (psi). This is the published minimum-hoop convention; a strength comparison and orientation model remain separate.
Before using the result
- State well orientation.
- Track each pore-pressure term.
- Compare stress with a separate strength criterion.
This is the published minimum-hoop convention; a strength comparison and orientation model remain separate.
Continue learning
Start with the broader workflow guide
Sources and method
The explanation and synthetic examples are original PetroCalcHub teaching material. The references below support the definitions and model context. They do not supply field validation for the synthetic scenarios.