{
  "version": 1,
  "topic": "WC04",
  "title": "Calculate hydrostatic pressure across two fluid columns",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Two static fluid columns",
    "premise": "Total vertical height 8,000 ft; upper fluid 9 ppg, lower fluid 12 ppg; zero top gauge pressure.",
    "formula": "ΔP = 0.052[MW1 h1 + MW2(8,000 − h1)]",
    "variables": "Each height is vertical and nonoverlapping. The fluid interface, not measured path length, divides the sum.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Upper vertical height (ft) = 2000",
        "explanation": "Total vertical height 8,000 ft; upper fluid 9 ppg, lower fluid 12 ppg; zero top gauge pressure."
      },
      {
        "label": "Select the relation",
        "calculation": "ΔP = 0.052[MW1 h1 + MW2(8,000 − h1)]",
        "explanation": "Each height is vertical and nonoverlapping. The fluid interface, not measured path length, divides the sum."
      },
      {
        "label": "Substitute the values",
        "calculation": "0.052 × 9 × 2,000 + 0.052 × 12 × 6,000 = 936 + 3,744 = 4,680 psi",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Upper head (psi) = 936; Lower head (psi) = 3744; Total head (psi) = 4680",
        "explanation": "Moving the lighter-fluid interface deeper reduces total static head at fixed total TVD."
      }
    ],
    "result": "Upper head (psi) = 936; Lower head (psi) = 3744; Total head (psi) = 4680",
    "interpretation": "Moving the lighter-fluid interface deeper reduces total static head at fixed total TVD.",
    "control": "Upper vertical height (ft)",
    "columns": [
      "Upper vertical height (ft)",
      "Upper head (psi)",
      "Lower head (psi)",
      "Total head (psi)"
    ],
    "rows": [
      [
        0,
        0,
        4992,
        4992
      ],
      [
        1000,
        468,
        4368,
        4836
      ],
      [
        2000,
        936,
        3744,
        4680
      ],
      [
        4000,
        1872,
        2496,
        4368
      ],
      [
        8000,
        3744,
        0,
        3744
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Upper vertical height (ft) = 0. Upper head (psi): 0; Lower head (psi): 4992; Total head (psi): 4992.",
      "Upper vertical height (ft) = 1000. Upper head (psi): 468; Lower head (psi): 4368; Total head (psi): 4836.",
      "Upper vertical height (ft) = 2000. Upper head (psi): 936; Lower head (psi): 3744; Total head (psi): 4680.",
      "Upper vertical height (ft) = 4000. Upper head (psi): 1872; Lower head (psi): 2496; Total head (psi): 4368.",
      "Upper vertical height (ft) = 8000. Upper head (psi): 3744; Lower head (psi): 0; Total head (psi): 3744."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Upper head (psi) without rounding intermediate values.",
        "answer": "0.052 × 9 × 2,000 + 0.052 × 12 × 6,000 = 936 + 3,744 = 4,680 psi. Upper head (psi) = 936. Moving the lighter-fluid interface deeper reduces total static head at fixed total TVD."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Upper vertical height (ft) changes from 0 to 8000. Other controls remain as stated in the premise. Upper head (psi) changes from 0 to 3744."
      },
      {
        "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. Each height is vertical and nonoverlapping. The fluid interface, not measured path length, divides the sum. Moving the lighter-fluid interface deeper reduces total static head at fixed total TVD."
      }
    ]
  },
  "recorded": null
}