{
  "version": 1,
  "topic": "PP08",
  "title": "Explore saturation curve parameters without changing the rock data",
  "createdAt": "2026-10-24",
  "case": {
    "title": "van Genuchten-Mualem Wetting Relative Permeability",
    "premise": "Fixed controls: S_e = 0.5 fraction. m is varied as shown.",
    "formula": "k_{rw}=\\sqrt{S_e}\\left[1-(1-S_e^{1/m})^m\\right]^2",
    "variables": "S_e: Effective Wetting Saturation (fraction). m: Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless).",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless) = 0.5",
        "explanation": "Fixed controls: S_e = 0.5 fraction. m is varied as shown."
      },
      {
        "label": "Select the relation",
        "calculation": "k_{rw}=\\sqrt{S_e}\\left[1-(1-S_e^{1/m})^m\\right]^2",
        "explanation": "S_e: Effective Wetting Saturation (fraction). m: Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless)."
      },
      {
        "label": "Substitute the values",
        "calculation": "S_e = 0.5; m = 0.5 → k_rw = 0.0126919957 fraction",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "k_rw (fraction) = 0.0126919957",
        "explanation": "Effective saturation remains 0.5; m changes curve shape under the linked model's m–n convention."
      }
    ],
    "result": "k_rw (fraction) = 0.0126919957",
    "interpretation": "Effective saturation remains 0.5; m changes curve shape under the linked model's m–n convention.",
    "control": "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless)",
    "columns": [
      "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless)",
      "k_rw (fraction)"
    ],
    "rows": [
      [
        0.2,
        0.000028329613137871383
      ],
      [
        0.35,
        0.0018141505297753593
      ],
      [
        0.5,
        0.012691995684869119
      ],
      [
        0.65,
        0.040690580332891586
      ],
      [
        0.8,
        0.08843215608467597
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless) = 0.2. k_rw (fraction): 0.0000283296131.",
      "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless) = 0.35. k_rw (fraction): 0.00181415053.",
      "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless) = 0.5. k_rw (fraction): 0.0126919957.",
      "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless) = 0.65. k_rw (fraction): 0.0406905803.",
      "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless) = 0.8. k_rw (fraction): 0.0884321561."
    ],
    "provenance": "Replayed with the same equation record and calculation engine as /calculators/van-genuchten-mualem-wetting-relative-permeability. Original synthetic inputs; engine outputs are saved in the downloadable record.",
    "questions": [
      {
        "question": "Reproduce the baseline k_rw (fraction) without rounding intermediate values.",
        "answer": "S_e = 0.5; m = 0.5 → k_rw = 0.0126919957 fraction. k_rw (fraction) = 0.0126919957. Effective saturation remains 0.5; m changes curve shape under the linked model's m–n convention."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless) changes from 0.2 to 0.8. Other controls remain as stated in the premise. k_rw (fraction) changes from 0.0000283296131 to 0.0884321561."
      },
      {
        "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. S_e: Effective Wetting Saturation (fraction). m: Shape Parameter (0 < m < 1; n = 1 / (1 - m)) (dimensionless). Effective saturation remains 0.5; m changes curve shape under the linked model's m–n convention."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "van-genuchten-mualem-wetting-relative-permeability",
    "parameter": "m",
    "inputs": {
      "S_e": "0.5",
      "m": "0.5"
    },
    "values": [
      0.2,
      0.35,
      0.5,
      0.65,
      0.8
    ],
    "outputs": [
      {
        "expression": "isZero(S_e) ? 0 : (isZero(1 - S_e) ? 1 : sqrt(S_e) * expm1(m * log1p(-S_e^(1 / m)))^2)",
        "intermediates": {},
        "inputs": {
          "S_e": 0.5,
          "m": 0.2
        },
        "target": "k_rw",
        "unit": "fraction",
        "value": 0.000028329613137871383
      },
      {
        "expression": "isZero(S_e) ? 0 : (isZero(1 - S_e) ? 1 : sqrt(S_e) * expm1(m * log1p(-S_e^(1 / m)))^2)",
        "intermediates": {},
        "inputs": {
          "S_e": 0.5,
          "m": 0.35
        },
        "target": "k_rw",
        "unit": "fraction",
        "value": 0.0018141505297753593
      },
      {
        "expression": "isZero(S_e) ? 0 : (isZero(1 - S_e) ? 1 : sqrt(S_e) * expm1(m * log1p(-S_e^(1 / m)))^2)",
        "intermediates": {},
        "inputs": {
          "S_e": 0.5,
          "m": 0.5
        },
        "target": "k_rw",
        "unit": "fraction",
        "value": 0.012691995684869119
      },
      {
        "expression": "isZero(S_e) ? 0 : (isZero(1 - S_e) ? 1 : sqrt(S_e) * expm1(m * log1p(-S_e^(1 / m)))^2)",
        "intermediates": {},
        "inputs": {
          "S_e": 0.5,
          "m": 0.65
        },
        "target": "k_rw",
        "unit": "fraction",
        "value": 0.040690580332891586
      },
      {
        "expression": "isZero(S_e) ? 0 : (isZero(1 - S_e) ? 1 : sqrt(S_e) * expm1(m * log1p(-S_e^(1 / m)))^2)",
        "intermediates": {},
        "inputs": {
          "S_e": 0.5,
          "m": 0.8
        },
        "target": "k_rw",
        "unit": "fraction",
        "value": 0.08843215608467597
      }
    ]
  }
}