{
  "version": 1,
  "topic": "RF10",
  "title": "The productivity gain from doubling permeability in a synthetic case",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Steady-State Radial Liquid Flow Rate",
    "premise": "Fixed controls: h = 50 ft; p_e = 3000 psi; p_wf = 2500 psi; mu_o = 2 cP; B_o = 1.2 bbl/STB; r_e = 1000 ft; r_w = 0.328 ft; s = 2 dimensionless. k 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": "Permeability (mD) = 100",
        "explanation": "Fixed controls: h = 50 ft; p_e = 3000 psi; p_wf = 2500 psi; mu_o = 2 cP; B_o = 1.2 bbl/STB; r_e = 1000 ft; r_w = 0.328 ft; s = 2 dimensionless. k 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": "Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis."
      }
    ],
    "result": "q_o (STB/day) = 735.844574",
    "interpretation": "Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis.",
    "control": "Permeability (mD)",
    "columns": [
      "Permeability (mD)",
      "q_o (STB/day)"
    ],
    "rows": [
      [
        25,
        183.96114354290125
      ],
      [
        50,
        367.9222870858025
      ],
      [
        100,
        735.844574171605
      ],
      [
        200,
        1471.68914834321
      ],
      [
        400,
        2943.37829668642
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Permeability (mD) = 25. q_o (STB/day): 183.961144.",
      "Permeability (mD) = 50. q_o (STB/day): 367.922287.",
      "Permeability (mD) = 100. q_o (STB/day): 735.844574.",
      "Permeability (mD) = 200. q_o (STB/day): 1471.68915.",
      "Permeability (mD) = 400. q_o (STB/day): 2943.3783."
    ],
    "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. Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Permeability (mD) changes from 25 to 400. Other controls remain as stated in the premise. q_o (STB/day) changes from 183.961144 to 2943.3783."
      },
      {
        "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). Stock-tank liquid rate changes linearly with k on the stated steady boundary and Bo basis."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "steady-state-radial-liquid-flow-rate",
    "parameter": "k",
    "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": [
      25,
      50,
      100,
      200,
      400
    ],
    "outputs": [
      {
        "expression": "(0.00708 * k * h * (p_e - p_wf)) / (mu_o * B_o * (ln(r_e / r_w) + s))",
        "intermediates": {},
        "inputs": {
          "k": 25,
          "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": 183.96114354290125
      },
      {
        "expression": "(0.00708 * k * h * (p_e - p_wf)) / (mu_o * B_o * (ln(r_e / r_w) + s))",
        "intermediates": {},
        "inputs": {
          "k": 50,
          "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": 367.9222870858025
      },
      {
        "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": 200,
          "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": 1471.68914834321
      },
      {
        "expression": "(0.00708 * k * h * (p_e - p_wf)) / (mu_o * B_o * (ln(r_e / r_w) + s))",
        "intermediates": {},
        "inputs": {
          "k": 400,
          "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": 2943.37829668642
      }
    ]
  }
}