{
  "version": 1,
  "topic": "GM04",
  "title": "Minimum and maximum hoop stress around a wellbore",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Minimum Hoop Stress at the Wellbore Wall",
    "premise": "Fixed controls: S_hmin = 6000 psi; S_hmax = 8000 psi; P_o = 3500 psi; S_dt = 300 psi. dP is varied as shown.",
    "formula": "sigma_theta_min = 3 * S_hmin - S_hmax - 2 * P_o - dP - S_dt",
    "variables": "S_hmin: Minimum Principal Stress in Reservoir (psi). S_hmax: Maximum Principal Stress in Reservoir (psi). P_o: Pore Pressure (psi). dP: Difference Between Wellbore Pressure and Pore Pressure (psi). S_dt: Thermally Induced Stress (psi).",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Difference Between Wellbore Pressure and Pore Pressure (psi) = 500",
        "explanation": "Fixed controls: S_hmin = 6000 psi; S_hmax = 8000 psi; P_o = 3500 psi; S_dt = 300 psi. dP is varied as shown."
      },
      {
        "label": "Select the relation",
        "calculation": "sigma_theta_min = 3 * S_hmin - S_hmax - 2 * P_o - dP - S_dt",
        "explanation": "S_hmin: Minimum Principal Stress in Reservoir (psi). S_hmax: Maximum Principal Stress in Reservoir (psi). P_o: Pore Pressure (psi). dP: Difference Between Wellbore Pressure and Pore Pressure (psi). S_dt: Thermally Induced Stress (psi)."
      },
      {
        "label": "Substitute the values",
        "calculation": "S_hmin = 6000; S_hmax = 8000; P_o = 3500; dP = 500; S_dt = 300 → sigma_theta_min = 2200 psi",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "sigma_theta_min (psi) = 2200",
        "explanation": "This is the published minimum-hoop convention; a strength comparison and orientation model remain separate."
      }
    ],
    "result": "sigma_theta_min (psi) = 2200",
    "interpretation": "This is the published minimum-hoop convention; a strength comparison and orientation model remain separate.",
    "control": "Difference Between Wellbore Pressure and Pore Pressure (psi)",
    "columns": [
      "Difference Between Wellbore Pressure and Pore Pressure (psi)",
      "sigma_theta_min (psi)"
    ],
    "rows": [
      [
        0,
        2700
      ],
      [
        250,
        2450
      ],
      [
        500,
        2200
      ],
      [
        750,
        1950
      ],
      [
        1000,
        1700
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Difference Between Wellbore Pressure and Pore Pressure (psi) = 0. sigma_theta_min (psi): 2700.",
      "Difference Between Wellbore Pressure and Pore Pressure (psi) = 250. sigma_theta_min (psi): 2450.",
      "Difference Between Wellbore Pressure and Pore Pressure (psi) = 500. sigma_theta_min (psi): 2200.",
      "Difference Between Wellbore Pressure and Pore Pressure (psi) = 750. sigma_theta_min (psi): 1950.",
      "Difference Between Wellbore Pressure and Pore Pressure (psi) = 1000. sigma_theta_min (psi): 1700."
    ],
    "provenance": "Replayed with the same equation record and calculation engine as /calculators/minimum-hoop-stress-at-the-wellbore-wall. Original synthetic inputs; engine outputs are saved in the downloadable record.",
    "questions": [
      {
        "question": "Reproduce the baseline sigma_theta_min (psi) without rounding intermediate values.",
        "answer": "S_hmin = 6000; S_hmax = 8000; P_o = 3500; dP = 500; S_dt = 300 → sigma_theta_min = 2200 psi. sigma_theta_min (psi) = 2200. This is the published minimum-hoop convention; a strength comparison and orientation model remain separate."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Difference Between Wellbore Pressure and Pore Pressure (psi) changes from 0 to 1000. Other controls remain as stated in the premise. sigma_theta_min (psi) changes from 2700 to 1700."
      },
      {
        "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_hmin: Minimum Principal Stress in Reservoir (psi). S_hmax: Maximum Principal Stress in Reservoir (psi). P_o: Pore Pressure (psi). dP: Difference Between Wellbore Pressure and Pore Pressure (psi). S_dt: Thermally Induced Stress (psi). This is the published minimum-hoop convention; a strength comparison and orientation model remain separate."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "minimum-hoop-stress-at-the-wellbore-wall",
    "parameter": "dP",
    "inputs": {
      "S_hmin": "6000",
      "S_hmax": "8000",
      "P_o": "3500",
      "dP": "500",
      "S_dt": "300"
    },
    "values": [
      0,
      250,
      500,
      750,
      1000
    ],
    "outputs": [
      {
        "expression": "(3 * S_hmin) - S_hmax - (2 * P_o) - dP - S_dt",
        "intermediates": {},
        "inputs": {
          "S_hmin": 6000,
          "S_hmax": 8000,
          "P_o": 3500,
          "dP": 0,
          "S_dt": 300
        },
        "target": "sigma_theta_min",
        "unit": "psi",
        "value": 2700
      },
      {
        "expression": "(3 * S_hmin) - S_hmax - (2 * P_o) - dP - S_dt",
        "intermediates": {},
        "inputs": {
          "S_hmin": 6000,
          "S_hmax": 8000,
          "P_o": 3500,
          "dP": 250,
          "S_dt": 300
        },
        "target": "sigma_theta_min",
        "unit": "psi",
        "value": 2450
      },
      {
        "expression": "(3 * S_hmin) - S_hmax - (2 * P_o) - dP - S_dt",
        "intermediates": {},
        "inputs": {
          "S_hmin": 6000,
          "S_hmax": 8000,
          "P_o": 3500,
          "dP": 500,
          "S_dt": 300
        },
        "target": "sigma_theta_min",
        "unit": "psi",
        "value": 2200
      },
      {
        "expression": "(3 * S_hmin) - S_hmax - (2 * P_o) - dP - S_dt",
        "intermediates": {},
        "inputs": {
          "S_hmin": 6000,
          "S_hmax": 8000,
          "P_o": 3500,
          "dP": 750,
          "S_dt": 300
        },
        "target": "sigma_theta_min",
        "unit": "psi",
        "value": 1950
      },
      {
        "expression": "(3 * S_hmin) - S_hmax - (2 * P_o) - dP - S_dt",
        "intermediates": {},
        "inputs": {
          "S_hmin": 6000,
          "S_hmax": 8000,
          "P_o": 3500,
          "dP": 1000,
          "S_dt": 300
        },
        "target": "sigma_theta_min",
        "unit": "psi",
        "value": 1700
      }
    ]
  }
}