{
  "version": 1,
  "topic": "PP12",
  "title": "Petrophysics units and saturation practice pack",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Archie Water Saturation from Resistivity Logs",
    "premise": "Fixed controls: a = 1 dimensionless; phi = 0.2 fraction; m = 2 dimensionless; R_w = 0.1 ohm m; n = 2 dimensionless. R_t is varied as shown.",
    "formula": "S_w = ((a / phi^m) * (R_w / R_t))^(1 / n)",
    "variables": "a: Tortuosity Factor (dimensionless). phi: Porosity (fraction). m: Cementation Factor (dimensionless). R_w: Formation Water Resistivity (ohm m). R_t: True Formation Resistivity (ohm m). n: Saturation Exponent (dimensionless).",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "True Formation Resistivity (ohm m) = 1",
        "explanation": "Fixed controls: a = 1 dimensionless; phi = 0.2 fraction; m = 2 dimensionless; R_w = 0.1 ohm m; n = 2 dimensionless. R_t is varied as shown."
      },
      {
        "label": "Select the relation",
        "calculation": "S_w = ((a / phi^m) * (R_w / R_t))^(1 / n)",
        "explanation": "a: Tortuosity Factor (dimensionless). phi: Porosity (fraction). m: Cementation Factor (dimensionless). R_w: Formation Water Resistivity (ohm m). R_t: True Formation Resistivity (ohm m). n: Saturation Exponent (dimensionless)."
      },
      {
        "label": "Substitute the values",
        "calculation": "a = 1; phi = 0.2; m = 2; R_w = 0.1; R_t = 1; n = 2 → S_w = 1.58113883 fraction",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "S_w (fraction) = 1.58113883",
        "explanation": "The full-water threshold is Rt = 2.5 Ω·m. Sw above one remains visible as an input/model inconsistency, not a physical saturation."
      }
    ],
    "result": "S_w (fraction) = 1.58113883",
    "interpretation": "The full-water threshold is Rt = 2.5 Ω·m. Sw above one remains visible as an input/model inconsistency, not a physical saturation.",
    "control": "True Formation Resistivity (ohm m)",
    "columns": [
      "True Formation Resistivity (ohm m)",
      "S_w (fraction)"
    ],
    "rows": [
      [
        1,
        1.5811388300841898
      ],
      [
        2,
        1.118033988749895
      ],
      [
        2.5,
        0.9999999999999999
      ],
      [
        5,
        0.7071067811865475
      ],
      [
        10,
        0.49999999999999994
      ]
    ],
    "baseline": 0,
    "rowNotes": [
      "True Formation Resistivity (ohm m) = 1. S_w (fraction): 1.58113883.",
      "True Formation Resistivity (ohm m) = 2. S_w (fraction): 1.11803399.",
      "True Formation Resistivity (ohm m) = 2.5. S_w (fraction): 1.",
      "True Formation Resistivity (ohm m) = 5. S_w (fraction): 0.707106781.",
      "True Formation Resistivity (ohm m) = 10. S_w (fraction): 0.5."
    ],
    "provenance": "Replayed with the same equation record and calculation engine as /calculators/archie-water-saturation-from-resistivity-logs. Original synthetic inputs; engine outputs are saved in the downloadable record.",
    "questions": [
      {
        "question": "Reproduce the baseline S_w (fraction) without rounding intermediate values.",
        "answer": "a = 1; phi = 0.2; m = 2; R_w = 0.1; R_t = 1; n = 2 → S_w = 1.58113883 fraction. S_w (fraction) = 1.58113883. The full-water threshold is Rt = 2.5 Ω·m. Sw above one remains visible as an input/model inconsistency, not a physical saturation."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "True Formation Resistivity (ohm m) changes from 1 to 10. Other controls remain as stated in the premise. S_w (fraction) changes from 1.58113883 to 0.5."
      },
      {
        "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. a: Tortuosity Factor (dimensionless). phi: Porosity (fraction). m: Cementation Factor (dimensionless). R_w: Formation Water Resistivity (ohm m). R_t: True Formation Resistivity (ohm m). n: Saturation Exponent (dimensionless). The full-water threshold is Rt = 2.5 Ω·m. Sw above one remains visible as an input/model inconsistency, not a physical saturation."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "archie-water-saturation-from-resistivity-logs",
    "parameter": "R_t",
    "inputs": {
      "a": "1",
      "phi": "0.2",
      "m": "2",
      "R_w": 0.1,
      "R_t": "20",
      "n": "2"
    },
    "values": [
      1,
      2,
      2.5,
      5,
      10
    ],
    "outputs": [
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.2,
          "m": 2,
          "R_w": 0.1,
          "R_t": 1,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 1.5811388300841898
      },
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.2,
          "m": 2,
          "R_w": 0.1,
          "R_t": 2,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 1.118033988749895
      },
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.2,
          "m": 2,
          "R_w": 0.1,
          "R_t": 2.5,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 0.9999999999999999
      },
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.2,
          "m": 2,
          "R_w": 0.1,
          "R_t": 5,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 0.7071067811865475
      },
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.2,
          "m": 2,
          "R_w": 0.1,
          "R_t": 10,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 0.49999999999999994
      }
    ]
  }
}