{
  "version": 1,
  "topic": "PP11",
  "title": "Brooks Corey versus van Genuchten on a shared synthetic case",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Normalized capillary-model comparison",
    "premise": "Brooks–Corey λ = 2 and van Genuchten m = 0.5. Both selected pressure scales are 5 psi; report Pc divided by that scale.",
    "formula": "Pc,BC/5 = Se^(−1/2); Pc,VG/5 = (Se^−2 − 1)^0.5",
    "variables": "The shared scale is a mathematical normalization, not evidence of identical measured entry pressure.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Effective saturation = 0.5",
        "explanation": "Brooks–Corey λ = 2 and van Genuchten m = 0.5. Both selected pressure scales are 5 psi; report Pc divided by that scale."
      },
      {
        "label": "Select the relation",
        "calculation": "Pc,BC/5 = Se^(−1/2); Pc,VG/5 = (Se^−2 − 1)^0.5",
        "explanation": "The shared scale is a mathematical normalization, not evidence of identical measured entry pressure."
      },
      {
        "label": "Substitute the values",
        "calculation": "At Se = 0.5: BC = 1/√0.5 = 1.41421356; VG = √(0.5^−2 − 1) = √3 = 1.73205081",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Brooks–Corey normalized Pc = 1.41421356; van Genuchten normalized Pc = 1.73205081",
        "explanation": "The shapes and endpoint limits differ; this is not a laboratory fit or accuracy ranking."
      }
    ],
    "result": "Brooks–Corey normalized Pc = 1.41421356; van Genuchten normalized Pc = 1.73205081",
    "interpretation": "The shapes and endpoint limits differ; this is not a laboratory fit or accuracy ranking.",
    "control": "Effective saturation",
    "columns": [
      "Effective saturation",
      "Brooks–Corey normalized Pc",
      "van Genuchten normalized Pc"
    ],
    "rows": [
      [
        0.1,
        3.162277660168379,
        9.949874371066198
      ],
      [
        0.25,
        2,
        3.872983346207417
      ],
      [
        0.5,
        1.414213562373095,
        1.7320508075688772
      ],
      [
        0.75,
        1.1547005383792515,
        0.8819171036881969
      ],
      [
        0.95,
        1.0259783520851542,
        0.3286841051788632
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Effective saturation = 0.1. Brooks–Corey normalized Pc: 3.16227766; van Genuchten normalized Pc: 9.94987437.",
      "Effective saturation = 0.25. Brooks–Corey normalized Pc: 2; van Genuchten normalized Pc: 3.87298335.",
      "Effective saturation = 0.5. Brooks–Corey normalized Pc: 1.41421356; van Genuchten normalized Pc: 1.73205081.",
      "Effective saturation = 0.75. Brooks–Corey normalized Pc: 1.15470054; van Genuchten normalized Pc: 0.881917104.",
      "Effective saturation = 0.95. Brooks–Corey normalized Pc: 1.02597835; van Genuchten normalized Pc: 0.328684105."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Brooks–Corey normalized Pc without rounding intermediate values.",
        "answer": "At Se = 0.5: BC = 1/√0.5 = 1.41421356; VG = √(0.5^−2 − 1) = √3 = 1.73205081. Brooks–Corey normalized Pc = 1.41421356. The shapes and endpoint limits differ; this is not a laboratory fit or accuracy ranking."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Effective saturation changes from 0.1 to 0.95. Other controls remain as stated in the premise. Brooks–Corey normalized Pc changes from 3.16227766 to 1.02597835."
      },
      {
        "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. The shared scale is a mathematical normalization, not evidence of identical measured entry pressure. The shapes and endpoint limits differ; this is not a laboratory fit or accuracy ranking."
      }
    ]
  },
  "recorded": null
}