{
  "version": 1,
  "topic": "DC10",
  "title": "Same first year different long term decline tails",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Matched first-year endpoint, different tails",
    "premise": "qi = 1,000 STB/day; all curves match 700 STB/day at day 365. Derived daily Di values for b 0, 0.5, 1: 0.000977191627, 0.0010697458, 0.0011741683.",
    "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; all curves match 700 STB/day at day 365. Derived daily Di values for b 0, 0.5, 1: 0.000977191627, 0.0010697458, 0.0011741683."
      },
      {
        "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 each derived Di makes q/qi = 0.7, so q = 700 STB/day.",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Exponential rate (STB/day) = 700; Hyperbolic b=0.5 rate (STB/day) = 700; Harmonic rate (STB/day) = 700",
        "explanation": "Matching one endpoint does not fix the intervening integral or the later tail."
      }
    ],
    "result": "Exponential rate (STB/day) = 700; Hyperbolic b=0.5 rate (STB/day) = 700; Harmonic rate (STB/day) = 700",
    "interpretation": "Matching one endpoint does not fix the intervening integral or the later tail.",
    "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
      ],
      [
        365,
        700,
        699.9999999999999,
        700
      ],
      [
        730,
        489.99999999999994,
        517.2312281263002,
        538.4615384615385
      ],
      [
        1825,
        168.06999999999994,
        256.07267770927365,
        318.18181818181813
      ],
      [
        3650,
        28.24752489999998,
        114.73161803801466,
        189.18918918918916
      ]
    ],
    "baseline": 1,
    "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) = 365. Exponential rate (STB/day): 700; Hyperbolic b=0.5 rate (STB/day): 700; Harmonic rate (STB/day): 700.",
      "Elapsed time (day) = 730. Exponential rate (STB/day): 490; Hyperbolic b=0.5 rate (STB/day): 517.231228; Harmonic rate (STB/day): 538.461538.",
      "Elapsed time (day) = 1825. Exponential rate (STB/day): 168.07; Hyperbolic b=0.5 rate (STB/day): 256.072678; Harmonic rate (STB/day): 318.181818.",
      "Elapsed time (day) = 3650. Exponential rate (STB/day): 28.2475249; Hyperbolic b=0.5 rate (STB/day): 114.731618; Harmonic rate (STB/day): 189.189189."
    ],
    "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 each derived Di makes q/qi = 0.7, so q = 700 STB/day.. Exponential rate (STB/day) = 700. Matching one endpoint does not fix the intervening integral or the later tail."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Elapsed time (day) changes from 0 to 3650. Other controls remain as stated in the premise. Exponential rate (STB/day) changes from 1000 to 28.2475249."
      },
      {
        "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. Matching one endpoint does not fix the intervening integral or the later tail."
      }
    ]
  },
  "recorded": null
}