{
  "version": 1,
  "topic": "PP02",
  "title": "Effective versus total porosity in saturation calculations",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Archie Water Saturation from Resistivity Logs",
    "premise": "Fixed controls: a = 1 dimensionless; m = 2 dimensionless; R_w = 0.05 ohm m; R_t = 20 ohm m; n = 2 dimensionless. phi 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": "Porosity (fraction) = 0.2",
        "explanation": "Fixed controls: a = 1 dimensionless; m = 2 dimensionless; R_w = 0.05 ohm m; R_t = 20 ohm m; n = 2 dimensionless. phi 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 = 0.05; R_t = 20; n = 2 → S_w = 0.25 fraction",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "S_w (fraction) = 0.25",
        "explanation": "Changing porosity with fixed exponents is a sensitivity; it does not determine the correct porosity definition."
      }
    ],
    "result": "S_w (fraction) = 0.25",
    "interpretation": "Changing porosity with fixed exponents is a sensitivity; it does not determine the correct porosity definition.",
    "control": "Porosity (fraction)",
    "columns": [
      "Porosity (fraction)",
      "S_w (fraction)"
    ],
    "rows": [
      [
        0.1,
        0.49999999999999994
      ],
      [
        0.15,
        0.3333333333333333
      ],
      [
        0.16,
        0.3125
      ],
      [
        0.2,
        0.24999999999999997
      ],
      [
        0.25,
        0.2
      ]
    ],
    "baseline": 3,
    "rowNotes": [
      "Porosity (fraction) = 0.1. S_w (fraction): 0.5.",
      "Porosity (fraction) = 0.15. S_w (fraction): 0.333333333.",
      "Porosity (fraction) = 0.16. S_w (fraction): 0.3125.",
      "Porosity (fraction) = 0.2. S_w (fraction): 0.25.",
      "Porosity (fraction) = 0.25. S_w (fraction): 0.2."
    ],
    "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 = 0.05; R_t = 20; n = 2 → S_w = 0.25 fraction. S_w (fraction) = 0.25. Changing porosity with fixed exponents is a sensitivity; it does not determine the correct porosity definition."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Porosity (fraction) changes from 0.1 to 0.25. Other controls remain as stated in the premise. S_w (fraction) changes from 0.5 to 0.2."
      },
      {
        "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). Changing porosity with fixed exponents is a sensitivity; it does not determine the correct porosity definition."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "archie-water-saturation-from-resistivity-logs",
    "parameter": "phi",
    "inputs": {
      "a": "1",
      "phi": "0.2",
      "m": "2",
      "R_w": "0.05",
      "R_t": "20",
      "n": "2"
    },
    "values": [
      0.1,
      0.15,
      0.16,
      0.2,
      0.25
    ],
    "outputs": [
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.1,
          "m": 2,
          "R_w": 0.05,
          "R_t": 20,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 0.49999999999999994
      },
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.15,
          "m": 2,
          "R_w": 0.05,
          "R_t": 20,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 0.3333333333333333
      },
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.16,
          "m": 2,
          "R_w": 0.05,
          "R_t": 20,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 0.3125
      },
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.2,
          "m": 2,
          "R_w": 0.05,
          "R_t": 20,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 0.24999999999999997
      },
      {
        "expression": "((a / (phi^m)) * (R_w / R_t))^(1 / n)",
        "intermediates": {},
        "inputs": {
          "a": 1,
          "phi": 0.25,
          "m": 2,
          "R_w": 0.05,
          "R_t": 20,
          "n": 2
        },
        "target": "S_w",
        "unit": "fraction",
        "value": 0.2
      }
    ]
  }
}