Herschel-Bulkley Rheology Fitting Calculator

Calculation sequence

1. Enter four to one hundred measured pairs with positive shear rates in 1/s and stress magnitudes in Pa. At least four distinct rates and a resolvable stress variation are required. Repeated-rate measurements are retained as repeated observations, not averaged or silently discarded. Preserve a single temperature, pressure, instrument geometry and sweep protocol; convert instrument speed and dial readings using the actual instrument calibration before entry.
2. The objective is unweighted least squares in shear-stress space: minimize the sum of squared Pa residuals. The implementation scales rates by their geometric mean and stresses by their maximum, then uses ML.js Levenberg-Marquardt with an analytical Jacobian and five deterministic starting exponents. Scaling changes numerical conditioning, not observation weighting. No logarithm of measured stress is used, so zero measured stress and noisy observations do not require a fabricated yield subtraction.
3. The software fits nonnegative yield stress and positive consistency with 0.1 <= n <= 2. Normalized stress coefficients have an upper bound of two. These are explicit software bounds, not universal rheological validity limits. Candidate fits must meet a projected-gradient or near-zero-error test; a returned optimizer array alone is not accepted as convergence. The best accepted start is selected, without claiming a proof of the global least-squares optimum.
4. Read the fitted parameters together with stress RMSE and the measured-versus-fitted residual table. RMSE is sqrt(sum of squared stress residuals / observation count), not a degrees-of-freedom-adjusted standard error. The prediction gives stress and apparent viscosity at the selected positive rate. Predictions outside the observed rate interval are explicitly flagged. Save and backup the measurements as well as the fitted outputs so a reopened case can be recalculated.

Worked check

The six default synthetic pairs follow tau = 10 + 0.2 sqrt(gamma): rates 1, 4, 25, 100, 400 and 900 1/s give 10.2, 10.4, 11, 12, 14 and 16 Pa. The fitter should recover tau_y = 10 Pa, K = 0.2 Pa.s^0.5 and n = 0.5 with essentially zero numerical RMSE. At 100 1/s, predicted stress is 12 Pa and apparent viscosity is 0.12 Pa.s. This exact recovery fixture does not represent field measurement precision.

Synthetic examples for reproducibility, not measured field results.

Assumptions and limitations

• A fitted Herschel-Bulkley yield stress is an extrapolated model intercept, not gel strength or an independently measured static yield point. No confidence intervals, outlier removal, measurement weighting, thixotropy, wall-slip correction or instrument-to-SI conversion is inferred.
• A good residual fit does not establish identifiable parameters. A rate span below one decade, a parameter on its bound or a zero yield estimate is reported for review. Singular values of the column-normalized fitting Jacobian supply a conditioning diagnostic: above 1e4 warns, above 1e12 rejects. These are software review thresholds, not uncertainty estimates. Bounds and finite/convergence checks reject unsupported cases rather than manufacturing a parameter set.
• Consistency has units Pa.s^n and is ordinary dynamic viscosity only when n = 1. Apparent viscosity is fitted stress divided by positive shear rate. The result is not an annular friction loss, ECD, turbulent friction factor or hydraulic design approval; those require their own geometry and flow model.

Sources and maintenance

Published by PetroCalcHub Team. Implementation and source checks: 2026-10-13. This does not claim independent expert review.

• USACE HEC-RAS mud/debris technical reference: Herschel-Bulkley parameters and consistency units
• Saasen and Ytrehus (2018): drilling-fluid fits, consistency units and shear-rate interval dependence
• ML.js Levenberg-Marquardt v5.1.0: bounded fitting and analytical Jacobian

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