{
  "version": 1,
  "topic": "DC02",
  "title": "Exponential hyperbolic and harmonic decline explained",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Matched initial rate and nominal decline",
    "premise": "qi = 1,000 STB/day and Di = 0.001/day for all curves. Only b changes among the three model columns.",
    "formula": "q = qi exp(−Dit) for b = 0; q = qi(1 + bDit)^(−1/b) otherwise",
    "variables": "Time in days, rate in STB/day. b = 1 is harmonic. These are synthetic model curves, not field fits.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Elapsed time (day) = 365",
        "explanation": "qi = 1,000 STB/day and Di = 0.001/day for all curves. Only b changes among the three model columns."
      },
      {
        "label": "Select the relation",
        "calculation": "q = qi exp(−Dit) for b = 0; q = qi(1 + bDit)^(−1/b) otherwise",
        "explanation": "Time in days, rate in STB/day. b = 1 is harmonic. These are synthetic model curves, not field fits."
      },
      {
        "label": "Substitute the values",
        "calculation": "At day 365: exponential = 1,000 exp(−0.365); hyperbolic = 1,000(1 + 0.5 × 0.365)^−2; harmonic = 1,000/1.365",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Exponential rate (STB/day) = 694.196651; Hyperbolic b=0.5 rate (STB/day) = 715.150919; Harmonic rate (STB/day) = 732.600733",
        "explanation": "Higher b sustains more late-time rate under these matched initial conditions."
      }
    ],
    "result": "Exponential rate (STB/day) = 694.196651; Hyperbolic b=0.5 rate (STB/day) = 715.150919; Harmonic rate (STB/day) = 732.600733",
    "interpretation": "Higher b sustains more late-time rate under these matched initial conditions.",
    "control": "Elapsed time (day)",
    "columns": [
      "Elapsed time (day)",
      "Exponential rate (STB/day)",
      "Hyperbolic b=0.5 rate (STB/day)",
      "Harmonic rate (STB/day)"
    ],
    "rows": [
      [
        0,
        1000,
        1000,
        1000
      ],
      [
        90,
        913.9311852712282,
        915.7299512373802,
        917.4311926605504
      ],
      [
        365,
        694.1966508779789,
        715.1509191924157,
        732.6007326007325
      ],
      [
        730,
        481.90899009020245,
        536.7038334071301,
        578.0346820809249
      ],
      [
        1825,
        161.21764412977677,
        273.3991199965825,
        353.98230088495575
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Elapsed time (day) = 0. Exponential rate (STB/day): 1000; Hyperbolic b=0.5 rate (STB/day): 1000; Harmonic rate (STB/day): 1000.",
      "Elapsed time (day) = 90. Exponential rate (STB/day): 913.931185; Hyperbolic b=0.5 rate (STB/day): 915.729951; Harmonic rate (STB/day): 917.431193.",
      "Elapsed time (day) = 365. Exponential rate (STB/day): 694.196651; Hyperbolic b=0.5 rate (STB/day): 715.150919; Harmonic rate (STB/day): 732.600733.",
      "Elapsed time (day) = 730. Exponential rate (STB/day): 481.90899; Hyperbolic b=0.5 rate (STB/day): 536.703833; Harmonic rate (STB/day): 578.034682.",
      "Elapsed time (day) = 1825. Exponential rate (STB/day): 161.217644; Hyperbolic b=0.5 rate (STB/day): 273.39912; Harmonic rate (STB/day): 353.982301."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Exponential rate (STB/day) without rounding intermediate values.",
        "answer": "At day 365: exponential = 1,000 exp(−0.365); hyperbolic = 1,000(1 + 0.5 × 0.365)^−2; harmonic = 1,000/1.365. Exponential rate (STB/day) = 694.196651. Higher b sustains more late-time rate under these matched initial conditions."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Elapsed time (day) changes from 0 to 1825. Other controls remain as stated in the premise. Exponential rate (STB/day) changes from 1000 to 161.217644."
      },
      {
        "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. Time in days, rate in STB/day. b = 1 is harmonic. These are synthetic model curves, not field fits. Higher b sustains more late-time rate under these matched initial conditions."
      }
    ]
  },
  "recorded": null
}