{
  "version": 1,
  "topic": "CE02",
  "title": "Annular capacity versus annular volume",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Capacity multiplied by measured length",
    "premise": "8.5 in hole around 5 in casing; the rounded library coefficient gives 0.0459 bbl/ft capacity.",
    "formula": "Va = 0.7854(Dh² − Do²)L/808.5",
    "variables": "Diameters in inches, measured length in feet, volume in petroleum barrels. Capacity is Va/L.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Measured length (ft) = 1000",
        "explanation": "8.5 in hole around 5 in casing; the rounded library coefficient gives 0.0459 bbl/ft capacity."
      },
      {
        "label": "Select the relation",
        "calculation": "Va = 0.7854(Dh² − Do²)L/808.5",
        "explanation": "Diameters in inches, measured length in feet, volume in petroleum barrels. Capacity is Va/L."
      },
      {
        "label": "Substitute the values",
        "calculation": "0.7854 × (8.5² − 5²) × 1,000/808.5 = 45.9 bbl",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Annular volume (bbl) = 45.9; Capacity (bbl/ft) = 0.0459",
        "explanation": "Length changes volume, while capacity is unchanged at fixed diameters."
      }
    ],
    "result": "Annular volume (bbl) = 45.9; Capacity (bbl/ft) = 0.0459",
    "interpretation": "Length changes volume, while capacity is unchanged at fixed diameters.",
    "control": "Measured length (ft)",
    "columns": [
      "Measured length (ft)",
      "Annular volume (bbl)",
      "Capacity (bbl/ft)"
    ],
    "rows": [
      [
        250,
        11.474999999999998,
        0.045899999999999996
      ],
      [
        500,
        22.949999999999996,
        0.045899999999999996
      ],
      [
        1000,
        45.89999999999999,
        0.045899999999999996
      ],
      [
        1500,
        68.85,
        0.045899999999999996
      ],
      [
        2000,
        91.79999999999998,
        0.045899999999999996
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Measured length (ft) = 250. Annular volume (bbl): 11.475; Capacity (bbl/ft): 0.0459.",
      "Measured length (ft) = 500. Annular volume (bbl): 22.95; Capacity (bbl/ft): 0.0459.",
      "Measured length (ft) = 1000. Annular volume (bbl): 45.9; Capacity (bbl/ft): 0.0459.",
      "Measured length (ft) = 1500. Annular volume (bbl): 68.85; Capacity (bbl/ft): 0.0459.",
      "Measured length (ft) = 2000. Annular volume (bbl): 91.8; Capacity (bbl/ft): 0.0459."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Annular volume (bbl) without rounding intermediate values.",
        "answer": "0.7854 × (8.5² − 5²) × 1,000/808.5 = 45.9 bbl. Annular volume (bbl) = 45.9. Length changes volume, while capacity is unchanged at fixed diameters."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Measured length (ft) changes from 250 to 2000. Other controls remain as stated in the premise. Annular volume (bbl) changes from 11.475 to 91.8."
      },
      {
        "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. Diameters in inches, measured length in feet, volume in petroleum barrels. Capacity is Va/L. Length changes volume, while capacity is unchanged at fixed diameters."
      }
    ]
  },
  "recorded": null
}