Herschel-Bulkley Annular Pressure Loss Calculator

Solve concentric laminar yield-power-law annular flow at a specified rate, retaining radial curvature, both yield surfaces and a moving plug.

Results

Calculate to view linked results. Default values are examples, not recommended operating inputs.

Learn, compare and practice

Worked explanations, controlled studies and printable exercises connected to this calculation.

Annular pressure loss through several hole sections Explore yield stress and the unyielded annular region

Calculation sequence

Use measured dynamic Herschel-Bulkley parameters at the modeled temperature, shear history and solids loading. K is in Pa.s^n, not a Newtonian viscosity in Pa.s; its dimensions change with n. The synthetic defaults are a numerical example, not a recommended mud. A rheology fit supplies parameters but does not verify laminar flow or concentric geometry.
For each trial pressure gradient the solver locates the two yield surfaces and matches the velocities obtained from the inner and outer stationary walls. Between these surfaces the liquid translates as an unsheared plug. The signed shear stress changes within the plug. This retains the cylindrical geometry rather than using a narrow-slot or equivalent-viscosity shortcut.
A second bracketed solve matches actual volume flow to the requested rate. Adaptive integration is repeated at tighter tolerance; requested-rate and two-wall residuals must pass before results appear. The reported ideal yield-onset gradient is a mathematical no-flow threshold, not a gel-breaking pressure or a transient restart prediction.
Read friction-only loss, the plug boundaries, wall shear rates and radial velocity profile together. The wall-viscosity inertia screening number uses rho*mean_velocity*(D-d) divided by the smallest apparent viscosity in the yielded layers. It is not a generalized Reynolds transition correlation. Confirm laminar behavior separately; CSV contains all 41 sampled radial stations.

Worked check

For D = 0.2 m, d = 0.1 m, Q = 0.001 m3/s, K = 0.2 Pa.s^n, n = 0.5, tau_y = 10 Pa and L = 1,000 m, the computed gradient is about 452.762896 Pa/m (onset 400 Pa/m). The plug extends from radius 0.052297760 to 0.096470991 m and translates at about 0.043803963 m/s. Friction loss is about 452,762.896 Pa. These synthetic values are independently checked using quadrature and bracketed roots, not field calibration.

Synthetic examples for reproducibility, not measured field results.

Assumptions and limitations

Incompressible, time-independent HB liquid; fully developed laminar axial flow in a stationary concentric circular annulus with no wall slip. No eccentricity, pipe rotation, turbulence, entrance losses, thixotropy, viscoelasticity, cuttings transport, gas or thermal/property variation is modeled.
Software envelopes: 0.05 <= d/D <= 0.95, 0.3 <= n <= 1, nonnegative yield stress, gradient <= 2,000,000 Pa/m and inertia screening number <= 1,000. These are conditioning/conservative implementation limits, not experimentally validated field ranges. A yield-stress/reference-stress ratio above 10,000 is rejected as unresolved near onset.
The output excludes hydrostatic head and all local losses. It is not total standpipe pressure, an ECD design or a hole-cleaning assessment. At tau_y = 0 and n = 1 it reduces to the Newtonian concentric solution; n < 1 and tau_y = 0 gives power-law flow. Density changes the screen, not the viscous gradient.

Implemented equations

Curved-annulus shear and yield surfaces

\begin{aligned}|\tau(r)|&=\frac{G_p}{2}\left|\frac{r_1r_2}{r}-r\right|\\r_2-r_1&=\frac{2\tau_y}{G_p}\\G_{\mathrm{onset}}&=\frac{2\tau_y}{R_o-R_i}\\\dot\gamma(r)&=\left[\frac{\max(0,|\tau(r)|-\tau_y)}{K}\right]^{1/n}\end{aligned}

Two-wall and volume-flow balances

\begin{aligned}\int_{R_i}^{r_1}\dot\gamma\,dr&=\int_{r_2}^{R_o}\dot\gamma\,dr\\Q&=2\pi\int_{R_i}^{R_o}ru(r)\,dr\\\Delta p_f&=G_pL\end{aligned}

Sources and maintenance

Published by PetroCalcHub Team. Implementation and source checks: 2026-10-14. This does not claim independent expert review.

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Related tools and references

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