{
  "version": 1,
  "topic": "RF05",
  "title": "Why well radius and drainage radius affect productivity differently",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Steady-State Radial Liquid Flow Rate",
    "premise": "Fixed controls: k = 100 mD; h = 50 ft; p_e = 3000 psi; p_wf = 2500 psi; mu_o = 2 cP; B_o = 1.2 bbl/STB; r_w = 0.328 ft; s = 2 dimensionless. r_e is varied as shown.",
    "formula": "q_o = 0.00708 k h (p_e - p_wf) / (mu_o B_o (ln(r_e / r_w) + s))",
    "variables": "k: Permeability (mD). h: Net Pay Thickness (ft). p_e: External Boundary Pressure (psi). p_wf: Flowing Bottom-Hole Pressure (psi). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). r_e: Drainage Radius (ft). r_w: Wellbore Radius (ft). s: Skin Factor (dimensionless).",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Drainage Radius (ft) = 1000",
        "explanation": "Fixed controls: k = 100 mD; h = 50 ft; p_e = 3000 psi; p_wf = 2500 psi; mu_o = 2 cP; B_o = 1.2 bbl/STB; r_w = 0.328 ft; s = 2 dimensionless. r_e is varied as shown."
      },
      {
        "label": "Select the relation",
        "calculation": "q_o = 0.00708 k h (p_e - p_wf) / (mu_o B_o (ln(r_e / r_w) + s))",
        "explanation": "k: Permeability (mD). h: Net Pay Thickness (ft). p_e: External Boundary Pressure (psi). p_wf: Flowing Bottom-Hole Pressure (psi). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). r_e: Drainage Radius (ft). r_w: Wellbore Radius (ft). s: Skin Factor (dimensionless)."
      },
      {
        "label": "Substitute the values",
        "calculation": "k = 100; h = 50; p_e = 3000; p_wf = 2500; mu_o = 2; B_o = 1.2; r_e = 1000; r_w = 0.328; s = 2 → q_o = 735.844574 STB/day",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "q_o (STB/day) = 735.844574",
        "explanation": "Increasing drainage radius increases logarithmic resistance under a maintained outer pressure."
      }
    ],
    "result": "q_o (STB/day) = 735.844574",
    "interpretation": "Increasing drainage radius increases logarithmic resistance under a maintained outer pressure.",
    "control": "Drainage Radius (ft)",
    "columns": [
      "Drainage Radius (ft)",
      "q_o (STB/day)"
    ],
    "rows": [
      [
        250,
        853.9632928313347
      ],
      [
        500,
        790.5159719158606
      ],
      [
        1000,
        735.844574171605
      ],
      [
        2000,
        688.2460737246969
      ],
      [
        4000,
        646.4313176701876
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Drainage Radius (ft) = 250. q_o (STB/day): 853.963293.",
      "Drainage Radius (ft) = 500. q_o (STB/day): 790.515972.",
      "Drainage Radius (ft) = 1000. q_o (STB/day): 735.844574.",
      "Drainage Radius (ft) = 2000. q_o (STB/day): 688.246074.",
      "Drainage Radius (ft) = 4000. q_o (STB/day): 646.431318."
    ],
    "provenance": "Replayed with the same equation record and calculation engine as /calculators/steady-state-radial-liquid-flow-rate. Original synthetic inputs; engine outputs are saved in the downloadable record.",
    "questions": [
      {
        "question": "Reproduce the baseline q_o (STB/day) without rounding intermediate values.",
        "answer": "k = 100; h = 50; p_e = 3000; p_wf = 2500; mu_o = 2; B_o = 1.2; r_e = 1000; r_w = 0.328; s = 2 → q_o = 735.844574 STB/day. q_o (STB/day) = 735.844574. Increasing drainage radius increases logarithmic resistance under a maintained outer pressure."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Drainage Radius (ft) changes from 250 to 4000. Other controls remain as stated in the premise. q_o (STB/day) changes from 853.963293 to 646.431318."
      },
      {
        "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. k: Permeability (mD). h: Net Pay Thickness (ft). p_e: External Boundary Pressure (psi). p_wf: Flowing Bottom-Hole Pressure (psi). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). r_e: Drainage Radius (ft). r_w: Wellbore Radius (ft). s: Skin Factor (dimensionless). Increasing drainage radius increases logarithmic resistance under a maintained outer pressure."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "steady-state-radial-liquid-flow-rate",
    "parameter": "r_e",
    "inputs": {
      "k": "100",
      "h": "50",
      "p_e": "3000",
      "p_wf": "2500",
      "mu_o": "2",
      "B_o": "1.2",
      "r_e": "1000",
      "r_w": "0.328",
      "s": "2"
    },
    "values": [
      250,
      500,
      1000,
      2000,
      4000
    ],
    "outputs": [
      {
        "expression": "(0.00708 * k * h * (p_e - p_wf)) / (mu_o * B_o * (ln(r_e / r_w) + s))",
        "intermediates": {},
        "inputs": {
          "k": 100,
          "h": 50,
          "p_e": 3000,
          "p_wf": 2500,
          "mu_o": 2,
          "B_o": 1.2,
          "r_e": 250,
          "r_w": 0.328,
          "s": 2
        },
        "target": "q_o",
        "unit": "STB/day",
        "value": 853.9632928313347
      },
      {
        "expression": "(0.00708 * k * h * (p_e - p_wf)) / (mu_o * B_o * (ln(r_e / r_w) + s))",
        "intermediates": {},
        "inputs": {
          "k": 100,
          "h": 50,
          "p_e": 3000,
          "p_wf": 2500,
          "mu_o": 2,
          "B_o": 1.2,
          "r_e": 500,
          "r_w": 0.328,
          "s": 2
        },
        "target": "q_o",
        "unit": "STB/day",
        "value": 790.5159719158606
      },
      {
        "expression": "(0.00708 * k * h * (p_e - p_wf)) / (mu_o * B_o * (ln(r_e / r_w) + s))",
        "intermediates": {},
        "inputs": {
          "k": 100,
          "h": 50,
          "p_e": 3000,
          "p_wf": 2500,
          "mu_o": 2,
          "B_o": 1.2,
          "r_e": 1000,
          "r_w": 0.328,
          "s": 2
        },
        "target": "q_o",
        "unit": "STB/day",
        "value": 735.844574171605
      },
      {
        "expression": "(0.00708 * k * h * (p_e - p_wf)) / (mu_o * B_o * (ln(r_e / r_w) + s))",
        "intermediates": {},
        "inputs": {
          "k": 100,
          "h": 50,
          "p_e": 3000,
          "p_wf": 2500,
          "mu_o": 2,
          "B_o": 1.2,
          "r_e": 2000,
          "r_w": 0.328,
          "s": 2
        },
        "target": "q_o",
        "unit": "STB/day",
        "value": 688.2460737246969
      },
      {
        "expression": "(0.00708 * k * h * (p_e - p_wf)) / (mu_o * B_o * (ln(r_e / r_w) + s))",
        "intermediates": {},
        "inputs": {
          "k": 100,
          "h": 50,
          "p_e": 3000,
          "p_wf": 2500,
          "mu_o": 2,
          "B_o": 1.2,
          "r_e": 4000,
          "r_w": 0.328,
          "s": 2
        },
        "target": "q_o",
        "unit": "STB/day",
        "value": 646.4313176701876
      }
    ]
  }
}