{
  "version": 1,
  "topic": "DC03",
  "title": "How the hyperbolic b factor changes a forecast",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Hyperbolic shape at a fixed horizon",
    "premise": "qi = 1,000 STB/day, Di = 0.001/day, horizon 365 days. Only b varies.",
    "formula": "q = qi(1 + bDit)^(−1/b); cumulative uses the matching Arps integral",
    "variables": "The b = 0 and b = 1 limits use their explicit exponential and logarithmic integrals.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Arps b = 0.5",
        "explanation": "qi = 1,000 STB/day, Di = 0.001/day, horizon 365 days. Only b varies."
      },
      {
        "label": "Select the relation",
        "calculation": "q = qi(1 + bDit)^(−1/b); cumulative uses the matching Arps integral",
        "explanation": "The b = 0 and b = 1 limits use their explicit exponential and logarithmic integrals."
      },
      {
        "label": "Substitute the values",
        "calculation": "At b = 0.5, q = 1,000(1 + 0.5 × 0.001 × 365)^−2; Np = 1,000/[0.001 × 0.5] × [1 − (1.1825)^−1]",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Rate at day 365 (STB/day) = 715.150919; Cumulative at day 365 (STB) = 308668.076",
        "explanation": "This is a deterministic shape sensitivity with no probability or reserves interpretation."
      }
    ],
    "result": "Rate at day 365 (STB/day) = 715.150919; Cumulative at day 365 (STB) = 308668.076",
    "interpretation": "This is a deterministic shape sensitivity with no probability or reserves interpretation.",
    "control": "Arps b",
    "columns": [
      "Arps b",
      "Rate at day 365 (STB/day)",
      "Cumulative at day 365 (STB)"
    ],
    "rows": [
      [
        0,
        694.1966508779789,
        305803.3491220212
      ],
      [
        0.25,
        705.1848494849675,
        307289.3773327054
      ],
      [
        0.5,
        715.1509191924157,
        308668.07610993675
      ],
      [
        0.75,
        724.2487216464871,
        309952.76321114897
      ],
      [
        1,
        732.6007326007325,
        311154.428636923
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Arps b = 0. Rate at day 365 (STB/day): 694.196651; Cumulative at day 365 (STB): 305803.349.",
      "Arps b = 0.25. Rate at day 365 (STB/day): 705.184849; Cumulative at day 365 (STB): 307289.377.",
      "Arps b = 0.5. Rate at day 365 (STB/day): 715.150919; Cumulative at day 365 (STB): 308668.076.",
      "Arps b = 0.75. Rate at day 365 (STB/day): 724.248722; Cumulative at day 365 (STB): 309952.763.",
      "Arps b = 1. Rate at day 365 (STB/day): 732.600733; Cumulative at day 365 (STB): 311154.429."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Rate at day 365 (STB/day) without rounding intermediate values.",
        "answer": "At b = 0.5, q = 1,000(1 + 0.5 × 0.001 × 365)^−2; Np = 1,000/[0.001 × 0.5] × [1 − (1.1825)^−1]. Rate at day 365 (STB/day) = 715.150919. This is a deterministic shape sensitivity with no probability or reserves interpretation."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Arps b changes from 0 to 1. Other controls remain as stated in the premise. Rate at day 365 (STB/day) changes from 694.196651 to 732.600733."
      },
      {
        "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. The b = 0 and b = 1 limits use their explicit exponential and logarithmic integrals. This is a deterministic shape sensitivity with no probability or reserves interpretation."
      }
    ]
  },
  "recorded": null
}