{
  "version": 1,
  "topic": "GM02",
  "title": "Effective stress versus total stress and the Biot coefficient",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Biot coefficient in a fixed-load sensitivity",
    "premise": "Compression positive; total normal stress 8,000 psi and pore pressure 3,000 psi. Only α varies.",
    "formula": "σ′ = σ − αp",
    "variables": "α is dimensionless. A coupled reservoir stress path could also change total stress and is not represented here.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Biot coefficient α = 0.8",
        "explanation": "Compression positive; total normal stress 8,000 psi and pore pressure 3,000 psi. Only α varies."
      },
      {
        "label": "Select the relation",
        "calculation": "σ′ = σ − αp",
        "explanation": "α is dimensionless. A coupled reservoir stress path could also change total stress and is not represented here."
      },
      {
        "label": "Substitute the values",
        "calculation": "8,000 − 0.8 × 3,000 = 5,600 psi",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Effective compression (psi) = 5600",
        "explanation": "Higher α gives a larger pore-pressure contribution at fixed total stress."
      }
    ],
    "result": "Effective compression (psi) = 5600",
    "interpretation": "Higher α gives a larger pore-pressure contribution at fixed total stress.",
    "control": "Biot coefficient α",
    "columns": [
      "Biot coefficient α",
      "Effective compression (psi)"
    ],
    "rows": [
      [
        0,
        8000
      ],
      [
        0.25,
        7250
      ],
      [
        0.5,
        6500
      ],
      [
        0.8,
        5600
      ],
      [
        1,
        5000
      ]
    ],
    "baseline": 3,
    "rowNotes": [
      "Biot coefficient α = 0. Effective compression (psi): 8000.",
      "Biot coefficient α = 0.25. Effective compression (psi): 7250.",
      "Biot coefficient α = 0.5. Effective compression (psi): 6500.",
      "Biot coefficient α = 0.8. Effective compression (psi): 5600.",
      "Biot coefficient α = 1. Effective compression (psi): 5000."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Effective compression (psi) without rounding intermediate values.",
        "answer": "8,000 − 0.8 × 3,000 = 5,600 psi. Effective compression (psi) = 5600. Higher α gives a larger pore-pressure contribution at fixed total stress."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Biot coefficient α changes from 0 to 1. Other controls remain as stated in the premise. Effective compression (psi) changes from 8000 to 5000."
      },
      {
        "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. α is dimensionless. A coupled reservoir stress path could also change total stress and is not represented here. Higher α gives a larger pore-pressure contribution at fixed total stress."
      }
    ]
  },
  "recorded": null
}