Engineering lesson
How to fit Herschel Bulkley parameters from viscometer readings
Fit Herschel–Bulkley to calibrated shear stress versus shear rate, not to unconverted dial readings. Instrument geometry, spring and rotor/bob configuration determine the conversion.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Convert measurements before fitting
A rotational speed is not automatically a shear rate, and a dial reading is not automatically pascals. Use the calibration and geometry for the specific instrument. The lesson therefore starts with an explicit SI table rather than assuming a universal RPM conversion.
The six synthetic points span 1–900 s⁻¹ and were generated from τy = 10 Pa, K = 0.2 Pa·s^0.5 and n = 0.5. They are exact teaching values, not instrument observations.
Inspect residuals after solving
A parameter estimate should be accompanied by predicted stresses and residuals at every measurement. A small aggregate error can hide a systematic low-shear bias.
The linked workflow performs bounded nonlinear fitting. Bounds, starting values and the measured range affect what the fitted parameters mean; a converged solution does not establish a unique physical yield stress.
Keep the data and intended use together
Retain the original readings, conversion details, temperature and conditioning history alongside the SI fit table. If the flow calculation operates outside the measured range, label that evaluation as extrapolation.
The explorer changes evaluation shear rate while preserving the fitted synthetic data. It does not recalibrate an instrument or create new laboratory evidence.
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.
The parameter set is fitted once. The table evaluates the resulting curve across measured and unmeasured shear rates.
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. Why should an RPM column not be relabelled s⁻¹ without checking the instrument?
Show the worked answer
The shear-rate conversion depends on geometry. Relabelling changes units without applying the necessary conversion.
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 instrument-specific calibration.
- Retain temperature and raw readings.
- Inspect low-shear residuals and extrapolation.
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.