Three models describe how a drilling fluid’s shear stress rises with shear rate, each a closer fit than the last. The Bingham plastic model draws a straight line through two readings. The power law curves that line to match shear-thinning behaviour. Herschel-Bulkley combines both — a yield stress plus a curve — and is the model API RP 13D builds its hydraulics on. Same viscometer, three different pictures of the same fluid.
Why a model at all
Plot shear stress against shear rate and a drilling fluid does not give the straight line through the origin that a Newtonian fluid would. It is shear-thinning and it has a yield stress — it resists flowing until pushed. A rheology model is just an equation that lets you predict the stress (and so the pressure and the carrying capacity) at shear rates you didn’t measure: down the drill pipe at thousands of reciprocal seconds, up the annulus at tens, at rest near zero.
Bingham plastic — the straight line
τ = YP + PV × γ. Two parameters from two readings: plastic viscosity (the slope, R600−R300) and yield point (the intercept, R300−PV). It is the workhorse of daily control because it is fast and the two numbers mean something physical. Its weakness is built into the straight line: fitted from the two high-shear points, it over-predicts the stress at the low shear rates that govern hole cleaning and the annulus. The Bingham YP is almost always higher than the fluid’s real yield stress.
Power law — the curve
τ = k × γn. Two parameters, no yield stress: the flow-behaviour index n = 3.32 log10(R600/R300) sets how shear-thinning the fluid is (n < 1), and the consistency index k = R300 / 511n sets its thickness. It fits the shear-thinning curve far better than a straight line — but it forces the stress to zero at zero shear rate, so it has no yield stress. For a mud that gels and suspends barite at rest, that is the wrong shape at the most important point.
Herschel-Bulkley — yield stress plus curve
τ = τ0 + k × γn. Three parameters: a true yield stress τ0 plus the power-law curve above it. It captures both the at-rest yield and the shear-thinning flow, which is why API RP 13D adopted it as the reference model for hydraulics and ECD prediction. The cost is that three parameters need more than two readings — you fit it from the low-shear points (3 and 6 rpm) together with 300 and 600, or from the full multi-speed curve.
Side by side
| Model | Equation | Parameters | Best for | Limitation |
|---|---|---|---|---|
| Bingham plastic | τ = τ0 + μpγ | PV, YP (2 reads) | Daily field control | Over-predicts low-shear stress |
| Power law | τ = kγn | n, k (2 reads) | Shear-thinning curve | No yield stress (wrong at rest) |
| Herschel-Bulkley | τ = τ0 + kγn | τ0, n, k (3+ reads) | Hydraulics, ECD, hole cleaning | Needs more readings / a fit |
Which one, when
For shift-to-shift control, read PV and YP — Bingham is enough to trend solids and carrying capacity. For anything that depends on what the fluid does at low shear — ECD, surge and swab, hole cleaning in a deviated well, barite suspension — use Herschel-Bulkley, because that is exactly where Bingham lies to you. The power law sits in between: better than a straight line, but blind to the yield stress that keeps your barite up.
Key takeaways
All three models start from the same viscometer. Bingham is two numbers and a straight line — fast, but it overstates low-shear stress. Power law curves the line but throws away the yield stress. Herschel-Bulkley keeps both and is what API RP 13D uses for hydraulics. Control daily with PV and YP; model the annulus with Herschel-Bulkley.