Engineering lesson
What pipe eccentricity changes in annular flow
An off-center pipe creates a narrow gap and a wide gap. Average annular area is unchanged, but velocity distribution, shear and the pressure needed to deliver a given rate can change.
By PetroCalcHub Team · Updated October 24, 2026 · Free to read and use
Equal area does not mean equal conductance
Moving the inner pipe sideways leaves the area difference unchanged until contact, but redistributes the gap. Flow preferentially uses the more open region, so Q/A cannot describe the local velocity profile.
The lesson uses the site's eccentric power-law model with all dimensions, rheology and rate fixed except the normalized eccentricity.
Identify the approximation behind the result
The linked workflow applies its documented eccentric-annulus closure; it is not a resolved rotating drillstring simulation. Its effective gap and flow-regime assumptions remain part of the result.
A lower calculated pressure loss does not establish improved hole cleaning. Low-velocity regions on the narrow side can still be important for deposited solids.
Check concentric and contact limits
At zero eccentricity the geometry should recover its concentric reference. Near contact, narrow-gap behavior makes geometric assumptions particularly consequential.
Do not infer actual eccentricity from pressure drop alone. Rheology, roughness, rate uncertainty and cuttings can produce similar changes in the measured pressure.
Explore the example
Original recorded synthetic cases solved with the site's eccentric-power-law-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.
Eccentricity (offset divided by radial clearance) (dimensionless) = 0.5. Eccentric annular friction pressure loss (Pa): 281776.86; Friction pressure gradient (Pa/m): 281.77686.
- Eccentric annular friction pressure loss (Pa)
Worked example
Eccentric Power-Law 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 = 0.03 m3/s; Measured power-law consistency = 0.2 Pa.s^n; Measured flow exponent = 0.6 dimensionless; Liquid density = 1200 kg/m3; Fully developed interval length = 1000 m; Absolute wall roughness = 0.00001 m.
Relation and unit basis
R = 1 - 0.072*(e/n)*(d/D)^0.8454 - 1.5*e^2*sqrt(n)*(d/D)^0.1852 + 0.96*e^3*sqrt(n)*(d/D)^0.2527. Dh=D-d. ReG=Deff^n*V^(2-n)*rho/[8^(n-1)*K*((3n+1)/(4n))^n].; fL=16/ReG. 1/sqrt(fT)=-4*log10(0.27*epsilon/Deff + 1.26^(n^-1.2)/(ReG*fT^(1-n/2))^(n^-0.75)). Select max(fL,fT). Loss=2*f*rho*V^2*L/Dh. These are Fanning factors; Darcy=4*f.
Eccentricity (offset divided by radial clearance) is varied within the supported input range. Power-law, time-independent incompressible liquid in a stationary uniform eccentric annulus. No yield stress/Herschel-Bulkley extrapolation, pipe rotation, cuttings bed, wall slip, gas, thermal feedback, local/entrance losses or hydrostatic head. This is friction-only, not a complete standpipe-pressure, ECD, hole-cleaning or well-control design.
1. Define the baseline
Eccentricity (offset divided by radial clearance) (dimensionless) = 0.5
Fixed controls: Outer bore inside diameter = 0.2 m; Inner pipe outside diameter = 0.1 m; Actual annular volume flow rate = 0.03 m3/s; Measured power-law consistency = 0.2 Pa.s^n; Measured flow exponent = 0.6 dimensionless; Liquid density = 1200 kg/m3; Fully developed interval length = 1000 m; Absolute wall roughness = 0.00001 m.
2. Select the relation
R = 1 - 0.072*(e/n)*(d/D)^0.8454 - 1.5*e^2*sqrt(n)*(d/D)^0.1852 + 0.96*e^3*sqrt(n)*(d/D)^0.2527. Dh=D-d. ReG=Deff^n*V^(2-n)*rho/[8^(n-1)*K*((3n+1)/(4n))^n].; fL=16/ReG. 1/sqrt(fT)=-4*log10(0.27*epsilon/Deff + 1.26^(n^-1.2)/(ReG*fT^(1-n/2))^(n^-0.75)). Select max(fL,fT). Loss=2*f*rho*V^2*L/Dh. These are Fanning factors; Darcy=4*f.
Eccentricity (offset divided by radial clearance) is varied within the supported input range. Power-law, time-independent incompressible liquid in a stationary uniform eccentric annulus. No yield stress/Herschel-Bulkley extrapolation, pipe rotation, cuttings bed, wall slip, gas, thermal feedback, local/entrance losses or hydrostatic head. This is friction-only, not a complete standpipe-pressure, ECD, hole-cleaning or well-control design.
3. Substitute the values
e = 0.5 dimensionless; with the fixed controls above, the workflow evaluates Eccentric annular friction pressure loss = 281776.86 Pa; Friction pressure gradient = 281.77686 Pa/m
Retain intermediate precision. The displayed values are rounded only after evaluation.
4. Read the result
Eccentric annular friction pressure loss (Pa) = 281776.86; Friction pressure gradient (Pa/m) = 281.77686
Retains the published correction's ranges: 0.3 <= d/D <= 0.9, 0.4 <= n <= 1 and 0 <= e <= 0.95. Pipe-wall contact at e=1 is deliberately excluded. Do not interpret the paper's approximately 5% laminar correction-fit statement as a 5% guarantee for turbulent pressure predictions or field muds. No fitted exponent is silently clamped. Additional software envelopes apply to every sweep point: 0.0001 <= ReG <= 10,000,000, epsilon/Deff <= 0.01 and gradient <= 2,000,000 Pa/m. The max-factor crossover is an empirical branch rule, not a resolution of mixed laminar/turbulent flow around the annulus. Use measured pressure-loss data to assess field applicability, particularly close to transition or for polymer elasticity.
Result and interpretation
Eccentric annular friction pressure loss (Pa) = 281776.86; Friction pressure gradient (Pa/m) = 281.77686
Retains the published correction's ranges: 0.3 <= d/D <= 0.9, 0.4 <= n <= 1 and 0 <= e <= 0.95. Pipe-wall contact at e=1 is deliberately excluded. Do not interpret the paper's approximately 5% laminar correction-fit statement as a 5% guarantee for turbulent pressure predictions or field muds. No fitted exponent is silently clamped. Additional software envelopes apply to every sweep point: 0.0001 <= ReG <= 10,000,000, epsilon/Deff <= 0.01 and gradient <= 2,000,000 Pa/m. The max-factor crossover is an empirical branch rule, not a resolution of mixed laminar/turbulent flow around the annulus. Use measured pressure-loss data to assess field applicability, particularly close to transition or for polymer elasticity.
Select eccentricity while retaining the same area and actual rate; the computed pressure response is model specific.
Exact scenario data
Original recorded synthetic cases solved with the site's eccentric-power-law-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.
| Eccentricity (offset divided by radial clearance) (dimensionless) | Eccentric annular friction pressure loss (Pa) | Friction pressure gradient (Pa/m) |
|---|---|---|
| 0 | 305,487 | 305.487 |
| 0.25 | 297,843.8 | 297.8438 |
| 0.5 | 281,776.9 | 281.7769 |
| 0.65 | 270,267 | 270.267 |
| 0.8 | 259,108.8 | 259.1088 |
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. Does average Q/A change solely because the pipe moves off center?
Show the worked answer
No, the cross-sectional area is unchanged. Local distribution and hydraulic resistance can still change.
2. Reproduce the baseline Eccentric annular friction pressure loss (Pa) without rounding intermediate values.
Show the worked answer
e = 0.5 dimensionless; with the fixed controls above, the workflow evaluates Eccentric annular friction pressure loss = 281776.86 Pa; Friction pressure gradient = 281.77686 Pa/m. Eccentric annular friction pressure loss (Pa) = 281776.86. Retains the published correction's ranges: 0.3 <= d/D <= 0.9, 0.4 <= n <= 1 and 0 <= e <= 0.95. Pipe-wall contact at e=1 is deliberately excluded. Do not interpret the paper's approximately 5% laminar correction-fit statement as a 5% guarantee for turbulent pressure predictions or field muds. No fitted exponent is silently clamped. Additional software envelopes apply to every sweep point: 0.0001 <= ReG <= 10,000,000, epsilon/Deff <= 0.01 and gradient <= 2,000,000 Pa/m. The max-factor crossover is an empirical branch rule, not a resolution of mixed laminar/turbulent flow around the annulus. Use measured pressure-loss data to assess field applicability, particularly close to transition or for polymer elasticity.
3. Compare the first and last recorded scenarios. Which input changes?
Show the worked answer
Eccentricity (offset divided by radial clearance) (dimensionless) changes from 0 to 0.8. Other controls remain as stated in the premise. Eccentric annular friction pressure loss (Pa) changes from 305487.015 to 259108.828.
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. Eccentricity (offset divided by radial clearance) is varied within the supported input range. Power-law, time-independent incompressible liquid in a stationary uniform eccentric annulus. No yield stress/Herschel-Bulkley extrapolation, pipe rotation, cuttings bed, wall slip, gas, thermal feedback, local/entrance losses or hydrostatic head. This is friction-only, not a complete standpipe-pressure, ECD, hole-cleaning or well-control design. Retains the published correction's ranges: 0.3 <= d/D <= 0.9, 0.4 <= n <= 1 and 0 <= e <= 0.95. Pipe-wall contact at e=1 is deliberately excluded. Do not interpret the paper's approximately 5% laminar correction-fit statement as a 5% guarantee for turbulent pressure predictions or field muds. No fitted exponent is silently clamped. Additional software envelopes apply to every sweep point: 0.0001 <= ReG <= 10,000,000, epsilon/Deff <= 0.01 and gradient <= 2,000,000 Pa/m. The max-factor crossover is an empirical branch rule, not a resolution of mixed laminar/turbulent flow around the annulus. Use measured pressure-loss data to assess field applicability, particularly close to transition or for polymer elasticity.
Before using the result
- State normalized eccentricity.
- Keep the same rate and rheology.
- Do not equate reduced loss with improved cleaning.
Retains the published correction's ranges: 0.3 <= d/D <= 0.9, 0.4 <= n <= 1 and 0 <= e <= 0.95. Pipe-wall contact at e=1 is deliberately excluded. Do not interpret the paper's approximately 5% laminar correction-fit statement as a 5% guarantee for turbulent pressure predictions or field muds. No fitted exponent is silently clamped. Additional software envelopes apply to every sweep point: 0.0001 <= ReG <= 10,000,000, epsilon/Deff <= 0.01 and gradient <= 2,000,000 Pa/m. The max-factor crossover is an empirical branch rule, not a resolution of mixed laminar/turbulent flow around the annulus. Use measured pressure-loss data to assess field applicability, particularly close to transition or for polymer elasticity.
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.