Engineering lesson
Bingham plastic versus power law versus Herschel Bulkley
Bingham plastic uses a yield stress plus a linear viscous term; power law uses a curved shear-rate response without a yield intercept; Herschel–Bulkley combines a yield term with a power-law term.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Compare models on the same measurements
The synthetic data follow τ = 10 + 0.2γ^0.5 in pascals. A Bingham line and a log-space power-law fit are fitted independently to those same points. The HB curve exactly reproduces the generating relation because it has the same structure.
That result is an internal synthetic check, not evidence that HB always fits field mud best. A fitted zero-shear intercept is also model dependent.
State the fitting objective
The Bingham comparison minimizes squared stress residuals. The power-law comparison minimizes squared log-stress residuals, changing the weighting. Their residual sums therefore cannot be ranked as if the fitting objectives were identical.
The linked workflow fits HB parameters only. It does not offer a three-model optimizer; the other two fits shown here are reproducible teaching calculations.
Check the shear-rate range
Consistency K has units Pa·s^n, so changing n changes its units as well as its value. Comparing bare K numbers across models is misleading.
A model selected over high shear rates may extrapolate poorly into a low-shear annulus. Plot residuals and evaluate the interval relevant to the intended flow calculation.
Explore the example
Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.
Select one of 6 recorded scenarios. Intermediate states are not interpolated.
Shear rate (1/s) = 100. HB generating stress (Pa): 12; Bingham predicted stress (Pa): 11.4049895; Power-law predicted stress (Pa): 12.8416811.
- HB generating stress (Pa)
- Bingham predicted stress (Pa)
- Power-law predicted stress (Pa)
Worked example
Shared synthetic rheology observations
Six exact SI observations generated by τ = 10 + 0.2√γ. Bingham is fitted in stress space; power law in log-stress space; HB uses the generating relation.
Relation and unit basis
τHB = 10 + 0.2√γ; τBingham = 10.7820903 + 0.00622899166γ; τpower = exp(2.26007752)γ^0.0635413424
Stress in Pa, shear rate in s⁻¹. The Bingham and power-law objectives differ; no common-objective accuracy ranking is implied.
1. Define the baseline
Shear rate (1/s) = 100
Six exact SI observations generated by τ = 10 + 0.2√γ. Bingham is fitted in stress space; power law in log-stress space; HB uses the generating relation.
2. Select the relation
τHB = 10 + 0.2√γ; τBingham = 10.7820903 + 0.00622899166γ; τpower = exp(2.26007752)γ^0.0635413424
Stress in Pa, shear rate in s⁻¹. The Bingham and power-law objectives differ; no common-objective accuracy ranking is implied.
3. Substitute the values
At γ = 100 s⁻¹: τHB = 10 + 0.2√100 = 12 Pa; Bingham uses the recorded fitted intercept and slope; power law uses exp(intercept) × 100^slope
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
HB generating stress (Pa) = 12; Bingham predicted stress (Pa) = 11.4049895; Power-law predicted stress (Pa) = 12.8416811
The HB workflow fits HB only; the other curves are independently reproduced teaching regressions.
Result and interpretation
HB generating stress (Pa) = 12; Bingham predicted stress (Pa) = 11.4049895; Power-law predicted stress (Pa) = 12.8416811
The HB workflow fits HB only; the other curves are independently reproduced teaching regressions.
Select a measured shear rate and compare predicted stresses and residuals from the three independently defined fits.
Exact scenario data
Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.
| Shear rate (1/s) | HB generating stress (Pa) | Bingham predicted stress (Pa) | Power-law predicted stress (Pa) |
|---|---|---|---|
| 1 | 10.2 | 10.78832 | 9.583832 |
| 4 | 10.4 | 10.80701 | 10.46634 |
| 25 | 11 | 10.93782 | 11.75889 |
| 100 | 12 | 11.40499 | 12.84168 |
| 400 | 14 | 13.27369 | 14.02418 |
| 900 | 16 | 16.38818 | 14.76576 |
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. Which HB choices recover the two simpler models?
Show the worked answer
n = 1 gives Bingham behavior; τy = 0 gives power-law behavior. Both together give a Newtonian relation.
2. Reproduce the baseline HB generating stress (Pa) without rounding intermediate values.
Show the worked answer
At γ = 100 s⁻¹: τHB = 10 + 0.2√100 = 12 Pa; Bingham uses the recorded fitted intercept and slope; power law uses exp(intercept) × 100^slope. HB generating stress (Pa) = 12. The HB workflow fits HB only; the other curves are independently reproduced teaching regressions.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Shear rate (1/s) changes from 1 to 900. Other controls remain as stated in the premise. HB generating stress (Pa) changes from 10.2 to 16.
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. Stress in Pa, shear rate in s⁻¹. The Bingham and power-law objectives differ; no common-objective accuracy ranking is implied. The HB workflow fits HB only; the other curves are independently reproduced teaching regressions.
Before using the result
- Use common measurements.
- Report each loss function.
- Do not claim the HB workflow fits all three models.
The HB workflow fits HB only; the other curves are independently reproduced teaching regressions.
Continue learning
Start with the broader workflow guide · Related method 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.