{ "version": 1, "topic": "DH07", "title": "What pipe eccentricity changes in annular flow", "createdAt": "2026-10-24", "case": { "title": "Eccentric Power-Law Annular Pressure Loss Calculator", "premise": "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.", "formula": "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.", "variables": "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.", "steps": [ { "label": "Define the baseline", "calculation": "Eccentricity (offset divided by radial clearance) (dimensionless) = 0.5", "explanation": "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." }, { "label": "Select the relation", "calculation": "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.", "explanation": "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." }, { "label": "Substitute the values", "calculation": "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", "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation." }, { "label": "Read the result", "calculation": "Eccentric annular friction pressure loss (Pa) = 281776.86; Friction pressure gradient (Pa/m) = 281.77686", "explanation": "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": "Eccentric annular friction pressure loss (Pa) = 281776.86; Friction pressure gradient (Pa/m) = 281.77686", "interpretation": "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.", "control": "Eccentricity (offset divided by radial clearance) (dimensionless)", "columns": [ "Eccentricity (offset divided by radial clearance) (dimensionless)", "Eccentric annular friction pressure loss (Pa)", "Friction pressure gradient (Pa/m)" ], "rows": [ [ 0, 305487.0151238772, 305.4870151238772 ], [ 0.25, 297843.83418669464, 297.84383418669466 ], [ 0.5, 281776.85983734136, 281.77685983734136 ], [ 0.65, 270267.0133654089, 270.2670133654089 ], [ 0.8, 259108.82813132525, 259.10882813132525 ] ], "baseline": 2, "rowNotes": [ "Eccentricity (offset divided by radial clearance) (dimensionless) = 0. Eccentric annular friction pressure loss (Pa): 305487.015; Friction pressure gradient (Pa/m): 305.487015.", "Eccentricity (offset divided by radial clearance) (dimensionless) = 0.25. Eccentric annular friction pressure loss (Pa): 297843.834; Friction pressure gradient (Pa/m): 297.843834.", "Eccentricity (offset divided by radial clearance) (dimensionless) = 0.5. Eccentric annular friction pressure loss (Pa): 281776.86; Friction pressure gradient (Pa/m): 281.77686.", "Eccentricity (offset divided by radial clearance) (dimensionless) = 0.65. Eccentric annular friction pressure loss (Pa): 270267.013; Friction pressure gradient (Pa/m): 270.267013.", "Eccentricity (offset divided by radial clearance) (dimensionless) = 0.8. Eccentric annular friction pressure loss (Pa): 259108.828; Friction pressure gradient (Pa/m): 259.108828." ], "provenance": "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.", "questions": [ { "question": "Reproduce the baseline Eccentric annular friction pressure loss (Pa) without rounding intermediate values.", "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." }, { "question": "Compare the first and last recorded scenarios. Which input changes?", "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." }, { "question": "What would be needed to apply this example to a different physical system?", "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." } ] }, "recorded": { "kind": "workflow", "workflow": "eccentric-power-law-annular-pressure-loss", "parameter": "e", "runs": [ { "inputs": { "D": 0.2, "d": 0.1, "Q": 0.03, "K": 0.2, "n": 0.6, "rho": 1200, "L": 1000, "e": 0, "epsilon": 0.00001 }, "result": { "outputs": [ { "label": "Eccentric annular friction pressure loss", "value": 305487.0151238772, "unit": "Pa" }, { "label": "Friction pressure gradient", "value": 305.4870151238772, "unit": "Pa/m" }, { "label": "Generalized Reynolds number", "value": 3400.8659997709124, "unit": "dimensionless" }, { "label": "Selected Fanning friction factor", "value": 0.00785165622120383, "unit": "dimensionless" }, { "label": "Darcy-equivalent friction factor", "value": 0.03140662488481532, "unit": "dimensionless" }, { "label": "Rheology-dependent effective diameter", "value": 0.06444070820236648, "unit": "m" }, { "label": 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This empirical branch selection is not a resolved mixed/transition flow field. The sweep must also remain inside the software envelope.", "Power-law liquid only: no yield stress, rotation, cuttings, gas, thermal coupling, hydrostatic head or local losses. The eccentricity correction is empirical, not an exact eccentric velocity field." ] } }, { "inputs": { "D": 0.2, "d": 0.1, "Q": 0.03, "K": 0.2, "n": 0.6, "rho": 1200, "L": 1000, "e": 0.25, "epsilon": 0.00001 }, "result": { "outputs": [ { "label": "Eccentric annular friction pressure loss", "value": 297843.83418669464, "unit": "Pa" }, { "label": "Friction pressure gradient", "value": 297.84383418669466, "unit": "Pa/m" }, { "label": "Generalized Reynolds number", "value": 3660.050719007908, "unit": "dimensionless" }, { "label": "Selected Fanning friction factor", "value": 0.007655210460224822, "unit": "dimensionless" }, { "label": "Darcy-equivalent friction factor", "value": 0.030620841840899287, "unit": "dimensionless" }, { "label": "Rheology-dependent effective diameter", "value": 0.0728321186602206, "unit": "m" }, { "label": "Laminar eccentric/concentric pressure ratio", "value": 0.929185484263658, "unit": "dimensionless" }, { "label": "Mean annular velocity", "value": 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