{ "version": 1, "topic": "FA02", "title": "Holdup versus flowing gas fraction", "createdAt": "2026-10-24", "case": { "title": "Vertical Drift-Flux Gas-Liquid Pressure Profile Calculator", "premise": "Fixed controls: Inlet absolute pressure = 5 MPa; Constant absolute temperature = 320 K; Pipe inside diameter = 0.1 m; Vertical upward interval length = 200 m; Gas mass flow rate = 0.05 kg/s; Liquid mass flow rate = 5 kg/s; Constant liquid density = 850 kg/m3; Gas molecular mass = 20 kg/kmol; Constant gas Z-factor = 0.9 dimensionless; Calibrated constant Darcy factor = 0.02 dimensionless; Calibrated distribution parameter = 1.2 dimensionless.", "formula": "jk = mass_rate_k/(A*rho_k); ug = C0*(jg+jl)+vd; alpha = jg/ug; ul = jl/(1-alpha); rhoh = alpha*rhog+(1-alpha)*rhol; rhog = p*M/(Z*8314.46261815324*T).; Pi = alpha*rhog*ug^2+(1-alpha)*rhol*ul^2; G = total_mass_rate/A; F = fD*G^2/(2*D*rhoh); -dp/ds = (F+rhoh*g)/(1+dPi/dp); acceleration loss = Pi_out-Pi_in. Darcy factor fD = 4*Fanning factor.", "variables": "Calibrated upward drift velocity is varied within the supported input range. Steady, vertical upward, co-current, isothermal gas/liquid screening only. No downward, counter-current, horizontal or inclined slip model; no regime selection, empirical Shi/Hagedorn-Brown/Beggs-Brill implementation, phase flash, dissolution, heat transfer, slug transients or reservoir coupling. This is a reduced model, not the full T2Well simulator.", "steps": [ { "label": "Define the baseline", "calculation": "Calibrated upward drift velocity (m/s) = 0.3", "explanation": "Fixed controls: Inlet absolute pressure = 5 MPa; Constant absolute temperature = 320 K; Pipe inside diameter = 0.1 m; Vertical upward interval length = 200 m; Gas mass flow rate = 0.05 kg/s; Liquid mass flow rate = 5 kg/s; Constant liquid density = 850 kg/m3; Gas molecular mass = 20 kg/kmol; Constant gas Z-factor = 0.9 dimensionless; Calibrated constant Darcy factor = 0.02 dimensionless; Calibrated distribution parameter = 1.2 dimensionless." }, { "label": "Select the relation", "calculation": "jk = mass_rate_k/(A*rho_k); ug = C0*(jg+jl)+vd; alpha = jg/ug; ul = jl/(1-alpha); rhoh = alpha*rhog+(1-alpha)*rhol; rhog = p*M/(Z*8314.46261815324*T).; Pi = alpha*rhog*ug^2+(1-alpha)*rhol*ul^2; G = total_mass_rate/A; F = fD*G^2/(2*D*rhoh); -dp/ds = (F+rhoh*g)/(1+dPi/dp); acceleration loss = Pi_out-Pi_in. Darcy factor fD = 4*Fanning factor.", "explanation": "Calibrated upward drift velocity is varied within the supported input range. Steady, vertical upward, co-current, isothermal gas/liquid screening only. No downward, counter-current, horizontal or inclined slip model; no regime selection, empirical Shi/Hagedorn-Brown/Beggs-Brill implementation, phase flash, dissolution, heat transfer, slug transients or reservoir coupling. This is a reduced model, not the full T2Well simulator." }, { "label": "Substitute the values", "calculation": "v_d = 0.3 m/s; with the fixed controls above, the workflow evaluates Outlet liquid holdup = 0.851738475 dimensionless; Drift-flux outlet absolute pressure = 3.52561111 MPa; No-slip comparison outlet pressure = 3.62785081 MPa", "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation." }, { "label": "Read the result", "calculation": "Outlet liquid holdup (dimensionless) = 0.851738475; Drift-flux outlet absolute pressure (MPa) = 3.52561111; No-slip comparison outlet pressure (MPa) = 3.62785081", "explanation": "C0, drift velocity, Z, Darcy factor and liquid density remain constant over the interval. The wall-friction closure is one specified mixture approximation, not a universal multiphase friction correlation. A Darcy factor must not be entered as a Fanning factor. Field use requires independent calibration and validation; numerical balance alone is not accuracy evidence. Software envelopes: 1 <= C0 <= 1.5, 0 <= drift velocity <= 3 m/s, positive rates for both phases, gas less dense than liquid, pressure >= 0.1 MPa absolute and 0 < gas void fraction <= 0.95 throughout. Both profiles must have acceleration parameter below 0.2; choking and near-critical solutions are rejected, not extrapolated. These are implementation limits, not universal correlation ranges." } ], "result": "Outlet liquid holdup (dimensionless) = 0.851738475; Drift-flux outlet absolute pressure (MPa) = 3.52561111; No-slip comparison outlet pressure (MPa) = 3.62785081", "interpretation": "C0, drift velocity, Z, Darcy factor and liquid density remain constant over the interval. The wall-friction closure is one specified mixture approximation, not a universal multiphase friction correlation. A Darcy factor must not be entered as a Fanning factor. Field use requires independent calibration and validation; numerical balance alone is not accuracy evidence. Software envelopes: 1 <= C0 <= 1.5, 0 <= drift velocity <= 3 m/s, positive rates for both phases, gas less dense than liquid, pressure >= 0.1 MPa absolute and 0 < gas void fraction <= 0.95 throughout. Both profiles must have acceleration parameter below 0.2; choking and near-critical solutions are rejected, not extrapolated. These are implementation limits, not universal correlation ranges.", "control": "Calibrated upward drift velocity (m/s)", "columns": [ "Calibrated upward drift velocity (m/s)", "Outlet liquid holdup (dimensionless)", "Drift-flux outlet absolute pressure (MPa)", "No-slip comparison outlet pressure (MPa)" ], "rows": [ [ 0, 0.8154814251695627, 3.578460530717927, 3.6278508073320284 ], [ 0.1, 0.829415235319943, 3.558005590328641, 3.6278508073320284 ], [ 0.3, 0.8517384751197616, 3.5256111059101793, 3.6278508073320284 ], [ 0.5, 0.8688516646342468, 3.5010993607529515, 3.6278508073320284 ], [ 0.7, 0.8823992097959685, 3.481897827300667, 3.6278508073320284 ] ], "baseline": 2, "rowNotes": [ "Calibrated upward drift velocity (m/s) = 0. Outlet liquid holdup (dimensionless): 0.815481425; Drift-flux outlet absolute pressure (MPa): 3.57846053; No-slip comparison outlet pressure (MPa): 3.62785081.", "Calibrated upward drift velocity (m/s) = 0.1. Outlet liquid holdup (dimensionless): 0.829415235; Drift-flux outlet absolute pressure (MPa): 3.55800559; No-slip comparison outlet pressure (MPa): 3.62785081.", "Calibrated upward drift velocity (m/s) = 0.3. Outlet liquid holdup (dimensionless): 0.851738475; Drift-flux outlet absolute pressure (MPa): 3.52561111; No-slip comparison outlet pressure (MPa): 3.62785081.", "Calibrated upward drift velocity (m/s) = 0.5. Outlet liquid holdup (dimensionless): 0.868851665; Drift-flux outlet absolute pressure (MPa): 3.50109936; No-slip comparison outlet pressure (MPa): 3.62785081.", "Calibrated upward drift velocity (m/s) = 0.7. Outlet liquid holdup (dimensionless): 0.88239921; Drift-flux outlet absolute pressure (MPa): 3.48189783; No-slip comparison outlet pressure (MPa): 3.62785081." ], "provenance": "Original recorded synthetic cases solved with the site's vertical-drift-flux-pressure-profile 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 Outlet liquid holdup (dimensionless) without rounding intermediate values.", "answer": "v_d = 0.3 m/s; with the fixed controls above, the workflow evaluates Outlet liquid holdup = 0.851738475 dimensionless; Drift-flux outlet absolute pressure = 3.52561111 MPa; No-slip comparison outlet pressure = 3.62785081 MPa. Outlet liquid holdup (dimensionless) = 0.851738475. C0, drift velocity, Z, Darcy factor and liquid density remain constant over the interval. The wall-friction closure is one specified mixture approximation, not a universal multiphase friction correlation. A Darcy factor must not be entered as a Fanning factor. Field use requires independent calibration and validation; numerical balance alone is not accuracy evidence. Software envelopes: 1 <= C0 <= 1.5, 0 <= drift velocity <= 3 m/s, positive rates for both phases, gas less dense than liquid, pressure >= 0.1 MPa absolute and 0 < gas void fraction <= 0.95 throughout. Both profiles must have acceleration parameter below 0.2; choking and near-critical solutions are rejected, not extrapolated. These are implementation limits, not universal correlation ranges." }, { "question": "Compare the first and last recorded scenarios. Which input changes?", "answer": "Calibrated upward drift velocity (m/s) changes from 0 to 0.7. Other controls remain as stated in the premise. Outlet liquid holdup (dimensionless) changes from 0.815481425 to 0.88239921." }, { "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. Calibrated upward drift velocity is varied within the supported input range. Steady, vertical upward, co-current, isothermal gas/liquid screening only. No downward, counter-current, horizontal or inclined slip model; no regime selection, empirical Shi/Hagedorn-Brown/Beggs-Brill implementation, phase flash, dissolution, heat transfer, slug transients or reservoir coupling. This is a reduced model, not the full T2Well simulator. C0, drift velocity, Z, Darcy factor and liquid density remain constant over the interval. The wall-friction closure is one specified mixture approximation, not a universal multiphase friction correlation. A Darcy factor must not be entered as a Fanning factor. Field use requires independent calibration and validation; numerical balance alone is not accuracy evidence. Software envelopes: 1 <= C0 <= 1.5, 0 <= drift velocity <= 3 m/s, positive rates for both phases, gas less dense than liquid, pressure >= 0.1 MPa absolute and 0 < gas void fraction <= 0.95 throughout. Both profiles must have acceleration parameter below 0.2; choking and near-critical solutions are rejected, not extrapolated. These are implementation limits, not universal correlation ranges." } ] }, "recorded": { "kind": "workflow", "workflow": "vertical-drift-flux-pressure-profile", "parameter": "v_d", "runs": [ { "inputs": { "p0": 5, "T": 320, "D": 0.1, "L": 200, "m_g": 0.05, "m_l": 5, "rho_l": 850, "M": 20, "Z": 0.9, "f_D": 0.02, "C_0": 1.2, "v_d": 0 }, "result": { "outputs": [ { "label": "Drift-flux outlet absolute pressure", "value": 3.578460530717927, "unit": "MPa" }, { "label": "No-slip comparison outlet pressure", "value": 3.6278508073320284, "unit": "MPa" }, { "label": "Total drift-flux pressure drop", "value": 1.4215394692820729, "unit": "MPa" }, { "label": "Integrated friction loss", "value": 0.011504312028795961, "unit": "MPa" }, { "label": "Integrated hydrostatic loss", "value": 1.410005028007209, "unit": "MPa" }, { "label": "Phase-acceleration pressure loss", "value": 0.000030129246073267837, "unit": "MPa" }, { "label": "Outlet liquid holdup", "value": 0.8154814251695627, "unit": "dimensionless" }, { "label": "Outlet gas velocity", "value": 1.1543578937811174, "unit": "m/s" }, { "label": "Outlet liquid velocity", "value": 0.918432247458948, "unit": "m/s" }, { "label": "Outlet gas/liquid velocity ratio", "value": 1.2568786614090603, "unit": "dimensionless" }, { "label": "Relative phase-momentum closure error", "value": 1.0446315036952147e-15, "unit": "dimensionless" } ], "columns": [ "Upward distance (m)", "Absolute pressure (MPa)", "Gas void fraction", "Liquid holdup", "Gas velocity (m/s)", "Liquid velocity (m/s)", "Holdup density (kg/m3)", "Friction loss (MPa)", "Hydrostatic loss (MPa)", "Acceleration loss (MPa)", "Acceleration parameter" ], "rows": [ [ 0, 5, 0.1409303495889832, 0.8590696504110168, 1.0816886346001666, 0.8718320309897688, 736.0946282111912, 0, 0, 0, 0.000015363624039267247 ], [ 4, 4.970911818684094, 0.14161488456596508, 0.858385115434035, 1.0827590899363038, 0.8725272894565513, 735.5069549334523, 0.0002247514487153846, 0.028862980437173127, 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No automatic regime/holdup correlation, phase flash, heat transfer or field validation is implied." ] } } ] } }