Engineering lesson
Explore yield stress and the unyielded annular region
A yield-stress fluid can contain a region with stress below the yield threshold. Such material may move as a plug; unyielded does not automatically mean stationary.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Use stress to identify yielding
In an ideal yield-stress constitutive law, deformation requires sufficient stress. A moving plug has nearly uniform velocity and little internal shear, while yielded regions deform around it.
The explorer varies yield stress in the exact laminar concentric HB model, holding rate, consistency and exponent fixed. It shows the solved radial velocity curve and inner and outer plug boundaries.
Distinguish a constitutive threshold from a flow map
The HB curve defines stress versus shear rate. The spatial location of a plug also depends on the momentum balance and geometry. A single τy value cannot specify plug radius in an arbitrary eccentric annulus.
This workflow solves the stationary concentric annulus with no-slip walls and independently checks flow rate and the two-wall velocity balance. Eccentricity, rotation and turbulence are outside this laminar profile.
Keep fitted parameters on their measurement basis
Yield stress can be sensitive to the fitting interval and conditioning history. Changing τy independently is useful for sensitivity, but it need not represent a physically realizable change in a particular mud.
Inspect the measured low-shear response before relying on an extrapolated yield intercept. A good high-shear fit alone may leave the plug-sensitive behavior uncertain.
Explore the example
Original recorded synthetic cases solved with the site's herschel-bulkley-annular-pressure-loss engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.
Select one of 5 recorded scenarios. Intermediate states are not interpolated.
Measured dynamic yield stress (Pa) = 5. Annular friction pressure loss (Pa): 243142.551; Inner plug boundary radius (m): 0.053535408; Outer plug boundary radius (m): 0.0946635444.
- Annular friction pressure loss (Pa)
Velocity and unyielded plug region
The shaded radial interval is unyielded in the stationary concentric laminar HB solution. It moves at the plug velocity while the adjoining regions shear.
Show radial velocity data
| Radius (m) | Velocity (m/s) |
|---|---|
| 0.05 | 0 |
| 0.05125 | 0.0327944 |
| 0.0525 | 0.0434758 |
| 0.05375 | 0.0445432 |
| 0.055 | 0.0445432 |
| 0.05625 | 0.0445432 |
| 0.0575 | 0.0445432 |
| 0.05875 | 0.0445432 |
| 0.06 | 0.0445432 |
| 0.06125 | 0.0445432 |
| 0.0625 | 0.0445432 |
| 0.06375 | 0.0445432 |
| 0.065 | 0.0445432 |
| 0.06625 | 0.0445432 |
| 0.0675 | 0.0445432 |
| 0.06875 | 0.0445432 |
| 0.07 | 0.0445432 |
| 0.07125 | 0.0445432 |
| 0.0725 | 0.0445432 |
| 0.07375 | 0.0445432 |
| 0.075 | 0.0445432 |
| 0.07625 | 0.0445432 |
| 0.0775 | 0.0445432 |
| 0.07875 | 0.0445432 |
| 0.08 | 0.0445432 |
| 0.08125 | 0.0445432 |
| 0.0825 | 0.0445432 |
| 0.08375 | 0.0445432 |
| 0.085 | 0.0445432 |
| 0.08625 | 0.0445432 |
| 0.0875 | 0.0445432 |
| 0.08875 | 0.0445432 |
| 0.09 | 0.0445432 |
| 0.09125 | 0.0445432 |
| 0.0925 | 0.0445432 |
| 0.09375 | 0.0445432 |
| 0.095 | 0.0445318 |
| 0.09625 | 0.0433487 |
| 0.0975 | 0.0377634 |
| 0.09875 | 0.0244072 |
| 0.1 | 0 |
Worked example
Herschel-Bulkley Annular Pressure Loss Calculator
Fixed controls: Outer bore inside diameter = 0.2 m; Inner pipe outside diameter = 0.1 m; Actual annular volume flow rate = 0.001 m3/s; Measured consistency factor = 0.2 Pa.s^n; Measured flow exponent = 0.5 dimensionless; Liquid density = 1200 kg/m3; Fully developed interval length = 1000 m.
Relation and unit basis
stress magnitude = Gp/2 * abs(r1*r2/r-r); r2-r1 = 2*tau_y/Gp; onset = 2*tau_y/(Ro-Ri); shear rate = (max(0,stress-tau_y)/K)^(1/n).; Inner and outer yielded-layer shear-rate integrals give the same plug velocity; Q = integral(2*pi*r*u(r), Ri..Ro); solve Gp for the requested Q; friction loss = Gp*L.
Measured dynamic yield stress is varied within the supported input range. 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.
1. Define the baseline
Measured dynamic yield stress (Pa) = 5
Fixed controls: Outer bore inside diameter = 0.2 m; Inner pipe outside diameter = 0.1 m; Actual annular volume flow rate = 0.001 m3/s; Measured consistency factor = 0.2 Pa.s^n; Measured flow exponent = 0.5 dimensionless; Liquid density = 1200 kg/m3; Fully developed interval length = 1000 m.
2. Select the relation
stress magnitude = Gp/2 * abs(r1*r2/r-r); r2-r1 = 2*tau_y/Gp; onset = 2*tau_y/(Ro-Ri); shear rate = (max(0,stress-tau_y)/K)^(1/n).; Inner and outer yielded-layer shear-rate integrals give the same plug velocity; Q = integral(2*pi*r*u(r), Ri..Ro); solve Gp for the requested Q; friction loss = Gp*L.
Measured dynamic yield stress is varied within the supported input range. 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.
3. Substitute the values
tau_y = 5 Pa; with the fixed controls above, the workflow evaluates Annular friction pressure loss = 243142.551 Pa; Inner plug boundary radius = 0.053535408 m; Outer plug boundary radius = 0.0946635444 m
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
Annular friction pressure loss (Pa) = 243142.551; Inner plug boundary radius (m) = 0.053535408; Outer plug boundary radius (m) = 0.0946635444
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.
Result and interpretation
Annular friction pressure loss (Pa) = 243142.551; Inner plug boundary radius (m) = 0.053535408; Outer plug boundary radius (m) = 0.0946635444
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.
The interaction changes a constitutive parameter, not the amount of solid material or a measured plug geometry.
Exact scenario data
Original recorded synthetic cases solved with the site's herschel-bulkley-annular-pressure-loss engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.
| Measured dynamic yield stress (Pa) | Annular friction pressure loss (Pa) | Inner plug boundary radius (m) | Outer plug boundary radius (m) |
|---|---|---|---|
| 0 | 20,694.73 | 0.07282895 | 0.07282895 |
| 1 | 69,055.09 | 0.05873797 | 0.08770035 |
| 2 | 113,907.5 | 0.05605435 | 0.09117057 |
| 5 | 243,142.6 | 0.05353541 | 0.09466354 |
| 10 | 452,762.9 | 0.05229776 | 0.09647099 |
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. Can an unyielded region travel down an annulus?
Show the worked answer
Yes. It can translate as a plug while adjoining yielded fluid deforms. Unyielded refers to deformation, not absolute velocity.
2. Reproduce the baseline Annular friction pressure loss (Pa) without rounding intermediate values.
Show the worked answer
tau_y = 5 Pa; with the fixed controls above, the workflow evaluates Annular friction pressure loss = 243142.551 Pa; Inner plug boundary radius = 0.053535408 m; Outer plug boundary radius = 0.0946635444 m. Annular friction pressure loss (Pa) = 243142.551. 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.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Measured dynamic yield stress (Pa) changes from 0 to 10. Other controls remain as stated in the premise. Annular friction pressure loss (Pa) changes from 20694.7344 to 452762.896.
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. Measured dynamic yield stress is varied within the supported input range. 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.
Before using the result
- Separate shear rate from translation speed.
- Read the workflow's closure limits.
- Keep τy tied to the fitted SI stress basis.
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.
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.