Original synthetic study
Why a good rheology fit can fail outside the measured range
A small residual over measured shear rates does not guarantee a good prediction outside that interval. Extrapolation tests assumptions that the fitted observations may not constrain.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Construct a deliberately narrow comparison
The synthetic HB curve supplies known stresses. A Bingham line fitted only to the two highest-shear points can match those points exactly while predicting a different low-shear response.
Exact agreement at two fitted points is expected for a two-parameter line. It gives no independent evidence about behavior near zero shear.
Separate fit error from extrapolation error
Residuals on the fitted range measure how well the selected objective matched those observations. Differences from the generating curve outside the range are an extrapolation diagnostic for this synthetic experiment.
A real mud has no known generating curve. Additional measurements at the intended shear rates are needed to assess the extrapolation rather than merely compare model shapes.
Preserve the fitting interval with the parameters
A parameter table without the measured range, temperature and stress units is difficult to reuse responsibly. The consistency unit also changes with exponent.
Use the HB fitting workflow to inspect residuals and evaluate a documented range. The study does not assert that the highest-order model will always perform best on independent data.
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): 12.8; Power-law predicted stress (Pa): 12.8416811.
- HB generating stress (Pa)
- Bingham predicted stress (Pa)
- Power-law predicted stress (Pa)
Worked example
A two-point high-shear fit extrapolated
Bingham line fitted only at 400 and 900 s⁻¹ from τ = 10 + 0.2√γ. Compare its low-shear extrapolation with the generating curve.
Relation and unit basis
τBingham = 12.4 + 0.004γ; τHB = 10 + 0.2√γ
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
Bingham line fitted only at 400 and 900 s⁻¹ from τ = 10 + 0.2√γ. Compare its low-shear extrapolation with the generating curve.
2. Select the relation
τBingham = 12.4 + 0.004γ; τHB = 10 + 0.2√γ
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) = 12.8; Power-law predicted stress (Pa) = 12.8416811
An exact two-point fit at high shear does not establish low-shear behavior.
Result and interpretation
HB generating stress (Pa) = 12; Bingham predicted stress (Pa) = 12.8; Power-law predicted stress (Pa) = 12.8416811
An exact two-point fit at high shear does not establish low-shear behavior.
Select a shear rate below or inside the two-point high-shear fit. The comparison makes the extrapolation gap visible.
Study data and reproduction
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 | 12.404 | 9.583832 |
| 4 | 10.4 | 12.416 | 10.46634 |
| 25 | 11 | 12.5 | 11.75889 |
| 100 | 12 | 12.8 | 12.84168 |
| 400 | 14 | 14 | 14.02418 |
| 900 | 16 | 16 | 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 does zero error at two points not establish a Bingham model for all shear rates?
Show the worked answer
Two parameters can interpolate two points exactly. Independent points and the relevant low-shear range are still untested.
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. An exact two-point fit at high shear does not establish low-shear behavior.
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. An exact two-point fit at high shear does not establish low-shear behavior.
Before using the result
- Record the measured shear range.
- Distinguish fitted from held-out points.
- Do not treat model-to-model agreement as laboratory validation.
An exact two-point fit at high shear does not establish low-shear behavior.
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.