{
  "version": 1,
  "topic": "PP09",
  "title": "Gassmann fluid substitution what the inputs represent",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Gassmann Saturated Rock Bulk Modulus",
    "premise": "Fixed controls: K_d = 10 GPa; K_s = 37 GPa; phi = 0.2 fraction. K_f is varied as shown.",
    "formula": "\\begin{aligned}K_{\\mathrm{sat}}&=K_d+\\frac{\\alpha_B^2}{\\phi/K_f+(\\alpha_B-\\phi)/K_s}\\\\\\alpha_B&=1-K_d/K_s\\end{aligned}",
    "variables": "K_d: Drained Rock-Frame Bulk Modulus (GPa). K_s: Solid Grain Bulk Modulus (GPa). K_f: Effective Pore-Fluid Bulk Modulus (GPa). phi: Connected Porosity (fraction).",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Effective Pore-Fluid Bulk Modulus (GPa) = 2.2",
        "explanation": "Fixed controls: K_d = 10 GPa; K_s = 37 GPa; phi = 0.2 fraction. K_f is varied as shown."
      },
      {
        "label": "Select the relation",
        "calculation": "\\begin{aligned}K_{\\mathrm{sat}}&=K_d+\\frac{\\alpha_B^2}{\\phi/K_f+(\\alpha_B-\\phi)/K_s}\\\\\\alpha_B&=1-K_d/K_s\\end{aligned}",
        "explanation": "K_d: Drained Rock-Frame Bulk Modulus (GPa). K_s: Solid Grain Bulk Modulus (GPa). K_f: Effective Pore-Fluid Bulk Modulus (GPa). phi: Connected Porosity (fraction)."
      },
      {
        "label": "Substitute the values",
        "calculation": "K_d = 10; K_s = 37; K_f = 2.2; phi = 0.2 → K_sat = 15.0605831 GPa",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "K_sat (GPa) = 15.0605831",
        "explanation": "The dry frame is fixed. Wave velocity additionally requires shear modulus and density."
      }
    ],
    "result": "K_sat (GPa) = 15.0605831",
    "interpretation": "The dry frame is fixed. Wave velocity additionally requires shear modulus and density.",
    "control": "Effective Pore-Fluid Bulk Modulus (GPa)",
    "columns": [
      "Effective Pore-Fluid Bulk Modulus (GPa)",
      "K_sat (GPa)"
    ],
    "rows": [
      [
        0.1,
        10.264360313315926
      ],
      [
        0.5,
        11.285260930888576
      ],
      [
        1,
        12.484662576687116
      ],
      [
        2.2,
        15.060583112457401
      ],
      [
        3,
        16.57546602525556
      ]
    ],
    "baseline": 3,
    "rowNotes": [
      "Effective Pore-Fluid Bulk Modulus (GPa) = 0.1. K_sat (GPa): 10.2643603.",
      "Effective Pore-Fluid Bulk Modulus (GPa) = 0.5. K_sat (GPa): 11.2852609.",
      "Effective Pore-Fluid Bulk Modulus (GPa) = 1. K_sat (GPa): 12.4846626.",
      "Effective Pore-Fluid Bulk Modulus (GPa) = 2.2. K_sat (GPa): 15.0605831.",
      "Effective Pore-Fluid Bulk Modulus (GPa) = 3. K_sat (GPa): 16.575466."
    ],
    "provenance": "Replayed with the same equation record and calculation engine as /calculators/gassmann-saturated-rock-bulk-modulus. Original synthetic inputs; engine outputs are saved in the downloadable record.",
    "questions": [
      {
        "question": "Reproduce the baseline K_sat (GPa) without rounding intermediate values.",
        "answer": "K_d = 10; K_s = 37; K_f = 2.2; phi = 0.2 → K_sat = 15.0605831 GPa. K_sat (GPa) = 15.0605831. The dry frame is fixed. Wave velocity additionally requires shear modulus and density."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Effective Pore-Fluid Bulk Modulus (GPa) changes from 0.1 to 3. Other controls remain as stated in the premise. K_sat (GPa) changes from 10.2643603 to 16.575466."
      },
      {
        "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. K_d: Drained Rock-Frame Bulk Modulus (GPa). K_s: Solid Grain Bulk Modulus (GPa). K_f: Effective Pore-Fluid Bulk Modulus (GPa). phi: Connected Porosity (fraction). The dry frame is fixed. Wave velocity additionally requires shear modulus and density."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "gassmann-saturated-rock-bulk-modulus",
    "parameter": "K_f",
    "inputs": {
      "K_d": "10",
      "K_s": "37",
      "K_f": "2.2",
      "phi": "0.2"
    },
    "values": [
      0.1,
      0.5,
      1,
      2.2,
      3
    ],
    "outputs": [
      {
        "expression": "K_d + (1 - K_d / K_s)^2 / (phi / K_f + (1 - phi) / K_s - K_d / K_s^2)",
        "intermediates": {},
        "inputs": {
          "K_d": 10,
          "K_s": 37,
          "K_f": 0.1,
          "phi": 0.2
        },
        "target": "K_sat",
        "unit": "GPa",
        "value": 10.264360313315926
      },
      {
        "expression": "K_d + (1 - K_d / K_s)^2 / (phi / K_f + (1 - phi) / K_s - K_d / K_s^2)",
        "intermediates": {},
        "inputs": {
          "K_d": 10,
          "K_s": 37,
          "K_f": 0.5,
          "phi": 0.2
        },
        "target": "K_sat",
        "unit": "GPa",
        "value": 11.285260930888576
      },
      {
        "expression": "K_d + (1 - K_d / K_s)^2 / (phi / K_f + (1 - phi) / K_s - K_d / K_s^2)",
        "intermediates": {},
        "inputs": {
          "K_d": 10,
          "K_s": 37,
          "K_f": 1,
          "phi": 0.2
        },
        "target": "K_sat",
        "unit": "GPa",
        "value": 12.484662576687116
      },
      {
        "expression": "K_d + (1 - K_d / K_s)^2 / (phi / K_f + (1 - phi) / K_s - K_d / K_s^2)",
        "intermediates": {},
        "inputs": {
          "K_d": 10,
          "K_s": 37,
          "K_f": 2.2,
          "phi": 0.2
        },
        "target": "K_sat",
        "unit": "GPa",
        "value": 15.060583112457401
      },
      {
        "expression": "K_d + (1 - K_d / K_s)^2 / (phi / K_f + (1 - phi) / K_s - K_d / K_s^2)",
        "intermediates": {},
        "inputs": {
          "K_d": 10,
          "K_s": 37,
          "K_f": 3,
          "phi": 0.2
        },
        "target": "K_sat",
        "unit": "GPa",
        "value": 16.57546602525556
      }
    ]
  }
}