{
  "version": 1,
  "topic": "CE05",
  "title": "How to combine casing open hole and shoe track volumes",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Two annuli and one internal shoe track",
    "premise": "Annulus A: 8.5/5 in × 1,000 ft; B: 7.75/5 in × 1,000 ft. Shoe-track ID 4.276 in. No excess is added in this geometry-only ledger.",
    "formula": "Vtotal = VA + VB + π ID² Ltrack/(4 × 808.5)",
    "variables": "Annular entries use the library's rounded 0.7854 coefficient; the internal track uses π/4. All lengths are measured feet.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Shoe-track length (ft) = 40",
        "explanation": "Annulus A: 8.5/5 in × 1,000 ft; B: 7.75/5 in × 1,000 ft. Shoe-track ID 4.276 in. No excess is added in this geometry-only ledger."
      },
      {
        "label": "Select the relation",
        "calculation": "Vtotal = VA + VB + π ID² Ltrack/(4 × 808.5)",
        "explanation": "Annular entries use the library's rounded 0.7854 coefficient; the internal track uses π/4. All lengths are measured feet."
      },
      {
        "label": "Substitute the values",
        "calculation": "VA = 45.9; VB = 34.0607143; track = π/4 × 4.276² × 40/808.5 bbl; add the three nonoverlapping entries",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Track volume (bbl) = 0.710469177; Total geometry volume (bbl) = 80.6711835",
        "explanation": "The internal track requires ID, while each annulus uses outer-boundary ID and inner-string OD."
      }
    ],
    "result": "Track volume (bbl) = 0.710469177; Total geometry volume (bbl) = 80.6711835",
    "interpretation": "The internal track requires ID, while each annulus uses outer-boundary ID and inner-string OD.",
    "control": "Shoe-track length (ft)",
    "columns": [
      "Shoe-track length (ft)",
      "Track volume (bbl)",
      "Total geometry volume (bbl)"
    ],
    "rows": [
      [
        0,
        0,
        79.96071428571427
      ],
      [
        20,
        0.35523458873557706,
        80.31594887444986
      ],
      [
        40,
        0.7104691774711541,
        80.67118346318543
      ],
      [
        60,
        1.065703766206731,
        81.02641805192101
      ],
      [
        100,
        1.776172943677885,
        81.73688722939215
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Shoe-track length (ft) = 0. Track volume (bbl): 0; Total geometry volume (bbl): 79.9607143.",
      "Shoe-track length (ft) = 20. Track volume (bbl): 0.355234589; Total geometry volume (bbl): 80.3159489.",
      "Shoe-track length (ft) = 40. Track volume (bbl): 0.710469177; Total geometry volume (bbl): 80.6711835.",
      "Shoe-track length (ft) = 60. Track volume (bbl): 1.06570377; Total geometry volume (bbl): 81.0264181.",
      "Shoe-track length (ft) = 100. Track volume (bbl): 1.77617294; Total geometry volume (bbl): 81.7368872."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Track volume (bbl) without rounding intermediate values.",
        "answer": "VA = 45.9; VB = 34.0607143; track = π/4 × 4.276² × 40/808.5 bbl; add the three nonoverlapping entries. Track volume (bbl) = 0.710469177. The internal track requires ID, while each annulus uses outer-boundary ID and inner-string OD."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Shoe-track length (ft) changes from 0 to 100. Other controls remain as stated in the premise. Track volume (bbl) changes from 0 to 1.77617294."
      },
      {
        "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. Annular entries use the library's rounded 0.7854 coefficient; the internal track uses π/4. All lengths are measured feet. The internal track requires ID, while each annulus uses outer-boundary ID and inner-string OD."
      }
    ]
  },
  "recorded": null
}