{
  "version": 1,
  "topic": "FA10",
  "title": "The same wellbore with three synthetic thermal histories",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Formation Thermal History, Shut-In and Restart Calculator",
    "premise": "Fixed controls: Well/formation interface radius = 0.1 m; Distant insulated outer radius = 10 m; Formation thermal conductivity = 2.5 W/(m.K); Formation volumetric heat capacity = 2.5 MJ/(m^3.K); Wellbore thermal capacity per length = 50000 J/(m.K); Wellbore-to-formation conductance per length = 50 W/(m.K); On/restart net heat input per length = 100 W/m; Initial uniform absolute temperature = 300 K; Initial operation duration = 24 h; Restart duration = 24 h.",
    "formula": "Cylindrical finite volumes use exact logarithmic radial thermal resistances, rock heat capacity per annular cell and one lumped wellbore capacity per length. Piecewise constant net heat input has on/zero/restart segments. Advance each linear segment by a matrix exponential and retain cumulative heat for an independent storage balance.",
    "variables": "Shut-in duration is varied within the supported input range. One local axial segment, uniform initial state and constant properties. No axial advection, geothermal gradient, full circulating well profile, casing-layer resolution or phase change.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Shut-in duration (h) = 24",
        "explanation": "Fixed controls: Well/formation interface radius = 0.1 m; Distant insulated outer radius = 10 m; Formation thermal conductivity = 2.5 W/(m.K); Formation volumetric heat capacity = 2.5 MJ/(m^3.K); Wellbore thermal capacity per length = 50000 J/(m.K); Wellbore-to-formation conductance per length = 50 W/(m.K); On/restart net heat input per length = 100 W/m; Initial uniform absolute temperature = 300 K; Initial operation duration = 24 h; Restart duration = 24 h."
      },
      {
        "label": "Select the relation",
        "calculation": "Cylindrical finite volumes use exact logarithmic radial thermal resistances, rock heat capacity per annular cell and one lumped wellbore capacity per length. Piecewise constant net heat input has on/zero/restart segments. Advance each linear segment by a matrix exponential and retain cumulative heat for an independent storage balance.",
        "explanation": "Shut-in duration is varied within the supported input range. One local axial segment, uniform initial state and constant properties. No axial advection, geothermal gradient, full circulating well profile, casing-layer resolution or phase change."
      },
      {
        "label": "Substitute the values",
        "calculation": "off = 24 h; with the fixed controls above, the workflow evaluates Final wellbore temperature = 312.975423 K; Wellbore temperature at shut-in = 311.732495 K",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Final wellbore temperature (K) = 312.975423; Wellbore temperature at shut-in (K) = 311.732495",
        "explanation": "Prescribed net heat rate: positive heats, negative cools, and shut-in means zero external heat input, not zero wellbore-to-rock heat transfer. Steel/fluid inventory and effective conductance must be supplied consistently. Insulated distant finite domain guarded by a diffusion-length test. This is not an arbitrary-duration infinite-reservoir solution. Energy/grid checks do not establish field accuracy or replace a calibrated transient well model."
      }
    ],
    "result": "Final wellbore temperature (K) = 312.975423; Wellbore temperature at shut-in (K) = 311.732495",
    "interpretation": "Prescribed net heat rate: positive heats, negative cools, and shut-in means zero external heat input, not zero wellbore-to-rock heat transfer. Steel/fluid inventory and effective conductance must be supplied consistently. Insulated distant finite domain guarded by a diffusion-length test. This is not an arbitrary-duration infinite-reservoir solution. Energy/grid checks do not establish field accuracy or replace a calibrated transient well model.",
    "control": "Shut-in duration (h)",
    "columns": [
      "Shut-in duration (h)",
      "Final wellbore temperature (K)",
      "Wellbore temperature at shut-in (K)"
    ],
    "rows": [
      [
        6,
        313.51205542161136,
        311.7324947195393
      ],
      [
        12,
        313.2869627238363,
        311.7324947195393
      ],
      [
        24,
        312.97542319039115,
        311.7324947195393
      ],
      [
        48,
        312.6221924017358,
        311.7324947195393
      ],
      [
        72,
        312.42622428053176,
        311.7324947195393
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Shut-in duration (h) = 6. Final wellbore temperature (K): 313.512055; Wellbore temperature at shut-in (K): 311.732495.",
      "Shut-in duration (h) = 12. Final wellbore temperature (K): 313.286963; Wellbore temperature at shut-in (K): 311.732495.",
      "Shut-in duration (h) = 24. Final wellbore temperature (K): 312.975423; Wellbore temperature at shut-in (K): 311.732495.",
      "Shut-in duration (h) = 48. Final wellbore temperature (K): 312.622192; Wellbore temperature at shut-in (K): 311.732495.",
      "Shut-in duration (h) = 72. Final wellbore temperature (K): 312.426224; Wellbore temperature at shut-in (K): 311.732495."
    ],
    "provenance": "Original recorded synthetic cases solved with the site's formation-thermal-history engine. The full inputs, outputs and diagnostic rows are in the downloadable JSON. These are discrete solved states; the lesson does not interpolate a new solve.",
    "questions": [
      {
        "question": "Reproduce the baseline Final wellbore temperature (K) without rounding intermediate values.",
        "answer": "off = 24 h; with the fixed controls above, the workflow evaluates Final wellbore temperature = 312.975423 K; Wellbore temperature at shut-in = 311.732495 K. Final wellbore temperature (K) = 312.975423. Prescribed net heat rate: positive heats, negative cools, and shut-in means zero external heat input, not zero wellbore-to-rock heat transfer. Steel/fluid inventory and effective conductance must be supplied consistently. Insulated distant finite domain guarded by a diffusion-length test. This is not an arbitrary-duration infinite-reservoir solution. Energy/grid checks do not establish field accuracy or replace a calibrated transient well model."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Shut-in duration (h) changes from 6 to 72. Other controls remain as stated in the premise. Final wellbore temperature (K) changes from 313.512055 to 312.426224."
      },
      {
        "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. Shut-in duration is varied within the supported input range. One local axial segment, uniform initial state and constant properties. No axial advection, geothermal gradient, full circulating well profile, casing-layer resolution or phase change. Prescribed net heat rate: positive heats, negative cools, and shut-in means zero external heat input, not zero wellbore-to-rock heat transfer. Steel/fluid inventory and effective conductance must be supplied consistently. Insulated distant finite domain guarded by a diffusion-length test. This is not an arbitrary-duration infinite-reservoir solution. Energy/grid checks do not establish field accuracy or replace a calibrated transient well model."
      }
    ]
  },
  "recorded": {
    "kind": "workflow",
    "workflow": "formation-thermal-history",
    "parameter": "off",
    "runs": [
      {
        "inputs": {
          "rw": 0.1,
          "re": 10,
          "k": 2.5,
          "capacity": 2.5,
          "Cw": 50000,
          "H": 50,
          "q": 100,
          "T": 300,
          "on": 24,
          "off": 6,
          "restart": 24
        },
        "result": {
          "outputs": [
            {
              "label": "Final wellbore temperature",
              "value": 313.51205542161136,
              "unit": "K"
            },
            {
              "label": "Wellbore temperature at shut-in",
              "value": 311.7324947195393,
              "unit": "K"
            },
            {
              "label": "Wellbore temperature before restart",
              "value": 304.59189026227784,
              "unit": "K"
            },
            {
              "label": "Maximum radial refinement change",
              "value": 0.006605277346352523,
              "unit": "K difference"
            },
            {
              "label": "Maximum energy balance residual",
              "value": 2.4959444999694824e-7,
              "unit": "J/m"
            }
          ],
          "columns": [
            "Elapsed time (h)",
            "Wellbore temperature (K)",
            "First formation cell temperature (K)",
            "Formation temperature at 2rw (K)",
            "Applied heat (W/m)",
            "Cumulative heat (J/m)",
            "Energy residual (J/m)"
          ],
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          },
          "warnings": [
            "One axial well segment with lumped wellbore thermal storage, one supplied conductance per length and radial formation conduction. Constant properties, initially uniform temperature, on/shut-in/restart prescribed heat input; positive heat warms the segment. Outer boundary is insulated and forced far beyond the diffusion front. This is not a full axial circulating well, geothermal gradient, time-varying mass-flow model or phase-change calculation. Heat capacity includes the user-selected fluid/steel inventory per length. No thermal safety or field-validation claim."
          ]
        }
      },
      {
        "inputs": {
          "rw": 0.1,
          "re": 10,
          "k": 2.5,
          "capacity": 2.5,
          "Cw": 50000,
          "H": 50,
          "q": 100,
          "T": 300,
          "on": 24,
          "off": 12,
          "restart": 24
        },
        "result": {
          "outputs": [
            {
              "label": "Final wellbore temperature",
              "value": 313.2869627238363,
              "unit": "K"
            },
            {
              "label": "Wellbore temperature at shut-in",
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              "unit": "K"
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            },
            {
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              "unit": "K difference"
            },
            {
              "label": "Maximum energy balance residual",
              "value": 2.942979335784912e-7,
              "unit": "J/m"
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            "First formation cell temperature (K)",
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            "Applied heat (W/m)",
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          ]
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      },
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          "H": 50,
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          "T": 300,
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          "warnings": [
            "One axial well segment with lumped wellbore thermal storage, one supplied conductance per length and radial formation conduction. Constant properties, initially uniform temperature, on/shut-in/restart prescribed heat input; positive heat warms the segment. Outer boundary is insulated and forced far beyond the diffusion front. This is not a full axial circulating well, geothermal gradient, time-varying mass-flow model or phase-change calculation. Heat capacity includes the user-selected fluid/steel inventory per length. No thermal safety or field-validation claim."
          ]
        }
      },
      {
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          "re": 10,
          "k": 2.5,
          "capacity": 2.5,
          "Cw": 50000,
          "H": 50,
          "q": 100,
          "T": 300,
          "on": 24,
          "off": 48,
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        "result": {
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              "label": "Final wellbore temperature",
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          "warnings": [
            "One axial well segment with lumped wellbore thermal storage, one supplied conductance per length and radial formation conduction. Constant properties, initially uniform temperature, on/shut-in/restart prescribed heat input; positive heat warms the segment. Outer boundary is insulated and forced far beyond the diffusion front. This is not a full axial circulating well, geothermal gradient, time-varying mass-flow model or phase-change calculation. Heat capacity includes the user-selected fluid/steel inventory per length. No thermal safety or field-validation claim."
          ]
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      },
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          "k": 2.5,
          "capacity": 2.5,
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          "H": 50,
          "q": 100,
          "T": 300,
          "on": 24,
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        "result": {
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            ]
          },
          "warnings": [
            "One axial well segment with lumped wellbore thermal storage, one supplied conductance per length and radial formation conduction. Constant properties, initially uniform temperature, on/shut-in/restart prescribed heat input; positive heat warms the segment. Outer boundary is insulated and forced far beyond the diffusion front. This is not a full axial circulating well, geothermal gradient, time-varying mass-flow model or phase-change calculation. Heat capacity includes the user-selected fluid/steel inventory per length. No thermal safety or field-validation claim."
          ]
        }
      }
    ]
  }
}