{
  "version": 1,
  "topic": "DC06",
  "title": "Modified hyperbolic decline and the terminal decline switch",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Continuous hyperbolic-to-exponential switch",
    "premise": "qi = 1,000 STB/day, Di = 0.002/day, b = 0.8, terminal nominal decline 0.0005/day. Switch time = 1875 days.",
    "formula": "q = qi(1 + bDi min(t,ts))^(−1/b) exp[−Dt max(0,t−ts)]",
    "variables": "Both decline coefficients are nominal per day. The tail starts at the rate reached at ts.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Elapsed time (day) = 1875",
        "explanation": "qi = 1,000 STB/day, Di = 0.002/day, b = 0.8, terminal nominal decline 0.0005/day. Switch time = 1875 days."
      },
      {
        "label": "Select the relation",
        "calculation": "q = qi(1 + bDi min(t,ts))^(−1/b) exp[−Dt max(0,t−ts)]",
        "explanation": "Both decline coefficients are nominal per day. The tail starts at the rate reached at ts."
      },
      {
        "label": "Substitute the values",
        "calculation": "ts = (0.002/0.0005 − 1)/(0.8 × 0.002) = 1875 days; q(ts) = 1,000(1 + 0.8 × 0.002 × ts)^−1.25",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Modified rate (STB/day) = 176.776695; Unswitched hyperbolic rate (STB/day) = 176.776695",
        "explanation": "Rate is continuous at the switch; the later exponential tail no longer follows the unswitched hyperbolic decline."
      }
    ],
    "result": "Modified rate (STB/day) = 176.776695; Unswitched hyperbolic rate (STB/day) = 176.776695",
    "interpretation": "Rate is continuous at the switch; the later exponential tail no longer follows the unswitched hyperbolic decline.",
    "control": "Elapsed time (day)",
    "columns": [
      "Elapsed time (day)",
      "Modified rate (STB/day)",
      "Unswitched hyperbolic rate (STB/day)"
    ],
    "rows": [
      [
        0,
        1000,
        1000
      ],
      [
        1000,
        302.88887004241076,
        302.8888700424107
      ],
      [
        1874,
        176.86512343628473,
        176.86512343628468
      ],
      [
        1875,
        176.77669529663692,
        176.77669529663686
      ],
      [
        1876,
        176.6883290423931,
        176.68834670651663
      ],
      [
        3650,
        72.77607983986526,
        90.40238180387689
      ],
      [
        7300,
        11.732788140783619,
        41.792748462422715
      ]
    ],
    "baseline": 3,
    "rowNotes": [
      "Elapsed time (day) = 0. Modified rate (STB/day): 1000; Unswitched hyperbolic rate (STB/day): 1000.",
      "Elapsed time (day) = 1000. Modified rate (STB/day): 302.88887; Unswitched hyperbolic rate (STB/day): 302.88887.",
      "Elapsed time (day) = 1874. Modified rate (STB/day): 176.865123; Unswitched hyperbolic rate (STB/day): 176.865123.",
      "Elapsed time (day) = 1875. Modified rate (STB/day): 176.776695; Unswitched hyperbolic rate (STB/day): 176.776695.",
      "Elapsed time (day) = 1876. Modified rate (STB/day): 176.688329; Unswitched hyperbolic rate (STB/day): 176.688347.",
      "Elapsed time (day) = 3650. Modified rate (STB/day): 72.7760798; Unswitched hyperbolic rate (STB/day): 90.4023818.",
      "Elapsed time (day) = 7300. Modified rate (STB/day): 11.7327881; Unswitched hyperbolic rate (STB/day): 41.7927485."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Modified rate (STB/day) without rounding intermediate values.",
        "answer": "ts = (0.002/0.0005 − 1)/(0.8 × 0.002) = 1875 days; q(ts) = 1,000(1 + 0.8 × 0.002 × ts)^−1.25. Modified rate (STB/day) = 176.776695. Rate is continuous at the switch; the later exponential tail no longer follows the unswitched hyperbolic decline."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Elapsed time (day) changes from 0 to 7300. Other controls remain as stated in the premise. Modified rate (STB/day) changes from 1000 to 11.7327881."
      },
      {
        "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. Both decline coefficients are nominal per day. The tail starts at the rate reached at ts. Rate is continuous at the switch; the later exponential tail no longer follows the unswitched hyperbolic decline."
      }
    ]
  },
  "recorded": null
}