{
  "version": 1,
  "topic": "RF06",
  "title": "Horizontal versus vertical well productivity under stated assumptions",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Joshi Horizontal Well Productivity Index",
    "premise": "Fixed controls: k_h = 100 mD; h = 50 ft; L = 1000 ft; A_ac = 40 acres; r_w = 0.328 ft; mu_o = 2 cP; B_o = 1.2 bbl/STB; s = 0 dimensionless. k_v is varied as shown.",
    "formula": "J_h = 0.00708 k_h h / [mu_o B_o (ln R_h + beta h/L ln(beta h / 2 r_w) + s)]",
    "variables": "k_h: Horizontal Permeability (mD). k_v: Vertical Permeability (mD). h: Net Pay Thickness (ft). L: Horizontal Well Length (ft). A_ac: Drainage Area (acres). r_w: Wellbore Radius (ft). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). s: Skin Factor (dimensionless).",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Vertical Permeability (mD) = 10",
        "explanation": "Fixed controls: k_h = 100 mD; h = 50 ft; L = 1000 ft; A_ac = 40 acres; r_w = 0.328 ft; mu_o = 2 cP; B_o = 1.2 bbl/STB; s = 0 dimensionless. k_v is varied as shown."
      },
      {
        "label": "Select the relation",
        "calculation": "J_h = 0.00708 k_h h / [mu_o B_o (ln R_h + beta h/L ln(beta h / 2 r_w) + s)]",
        "explanation": "k_h: Horizontal Permeability (mD). k_v: Vertical Permeability (mD). h: Net Pay Thickness (ft). L: Horizontal Well Length (ft). A_ac: Drainage Area (acres). r_w: Wellbore Radius (ft). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). s: Skin Factor (dimensionless)."
      },
      {
        "label": "Substitute the values",
        "calculation": "k_h = 100; k_v = 10; h = 50; L = 1000; A_ac = 40; r_w = 0.328; mu_o = 2; B_o = 1.2; s = 0 → J_h = 7.5062339 STB/day/psi",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "J_h (STB/day/psi) = 7.5062339",
        "explanation": "This is Joshi horizontal PI with fixed kh and geometry; it is not a horizontal-to-vertical rate multiplier."
      }
    ],
    "result": "J_h (STB/day/psi) = 7.5062339",
    "interpretation": "This is Joshi horizontal PI with fixed kh and geometry; it is not a horizontal-to-vertical rate multiplier.",
    "control": "Vertical Permeability (mD)",
    "columns": [
      "Vertical Permeability (mD)",
      "J_h (STB/day/psi)"
    ],
    "rows": [
      [
        1,
        3.3402068170004964
      ],
      [
        5,
        6.1413493451923875
      ],
      [
        10,
        7.5062338989974755
      ],
      [
        25,
        9.216046240596464
      ],
      [
        100,
        11.221220168828001
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Vertical Permeability (mD) = 1. J_h (STB/day/psi): 3.34020682.",
      "Vertical Permeability (mD) = 5. J_h (STB/day/psi): 6.14134935.",
      "Vertical Permeability (mD) = 10. J_h (STB/day/psi): 7.5062339.",
      "Vertical Permeability (mD) = 25. J_h (STB/day/psi): 9.21604624.",
      "Vertical Permeability (mD) = 100. J_h (STB/day/psi): 11.2212202."
    ],
    "provenance": "Replayed with the same equation record and calculation engine as /calculators/joshi-horizontal-well-productivity-index. Original synthetic inputs; engine outputs are saved in the downloadable record.",
    "questions": [
      {
        "question": "Reproduce the baseline J_h (STB/day/psi) without rounding intermediate values.",
        "answer": "k_h = 100; k_v = 10; h = 50; L = 1000; A_ac = 40; r_w = 0.328; mu_o = 2; B_o = 1.2; s = 0 → J_h = 7.5062339 STB/day/psi. J_h (STB/day/psi) = 7.5062339. This is Joshi horizontal PI with fixed kh and geometry; it is not a horizontal-to-vertical rate multiplier."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Vertical Permeability (mD) changes from 1 to 100. Other controls remain as stated in the premise. J_h (STB/day/psi) changes from 3.34020682 to 11.2212202."
      },
      {
        "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_h: Horizontal Permeability (mD). k_v: Vertical Permeability (mD). h: Net Pay Thickness (ft). L: Horizontal Well Length (ft). A_ac: Drainage Area (acres). r_w: Wellbore Radius (ft). mu_o: Oil Viscosity (cP). B_o: Oil Formation Volume Factor (bbl/STB). s: Skin Factor (dimensionless). This is Joshi horizontal PI with fixed kh and geometry; it is not a horizontal-to-vertical rate multiplier."
      }
    ]
  },
  "recorded": {
    "kind": "scalar",
    "equation": "joshi-horizontal-well-productivity-index",
    "parameter": "k_v",
    "inputs": {
      "k_h": "100",
      "k_v": "10",
      "h": "50",
      "L": "1000",
      "A_ac": "40",
      "r_w": "0.328",
      "mu_o": "2",
      "B_o": "1.2",
      "s": "0"
    },
    "values": [
      1,
      5,
      10,
      25,
      100
    ],
    "outputs": [
      {
        "expression": "(0.00708 * k_h * h) / (mu_o * B_o * D_joshi)",
        "intermediates": {
          "A_ft2": 1742400,
          "r_eh": 744.7302502830383,
          "beta": 10,
          "a_joshi": 832.7873756978502,
          "R_h": 2.997542938886246,
          "D_joshi": 4.41589422694655
        },
        "inputs": {
          "k_h": 100,
          "k_v": 1,
          "h": 50,
          "L": 1000,
          "A_ac": 40,
          "r_w": 0.328,
          "mu_o": 2,
          "B_o": 1.2,
          "s": 0
        },
        "target": "J_h",
        "unit": "STB/day/psi",
        "value": 3.3402068170004964
      },
      {
        "expression": "(0.00708 * k_h * h) / (mu_o * B_o * D_joshi)",
        "intermediates": {
          "A_ft2": 1742400,
          "r_eh": 744.7302502830383,
          "beta": 4.47213595499958,
          "a_joshi": 832.7873756978502,
          "R_h": 2.997542938886246,
          "D_joshi": 2.4017523138537458
        },
        "inputs": {
          "k_h": 100,
          "k_v": 5,
          "h": 50,
          "L": 1000,
          "A_ac": 40,
          "r_w": 0.328,
          "mu_o": 2,
          "B_o": 1.2,
          "s": 0
        },
        "target": "J_h",
        "unit": "STB/day/psi",
        "value": 6.1413493451923875
      },
      {
        "expression": "(0.00708 * k_h * h) / (mu_o * B_o * D_joshi)",
        "intermediates": {
          "A_ft2": 1742400,
          "r_eh": 744.7302502830383,
          "beta": 3.1622776601683795,
          "a_joshi": 832.7873756978502,
          "R_h": 2.997542938886246,
          "D_joshi": 1.9650333574031045
        },
        "inputs": {
          "k_h": 100,
          "k_v": 10,
          "h": 50,
          "L": 1000,
          "A_ac": 40,
          "r_w": 0.328,
          "mu_o": 2,
          "B_o": 1.2,
          "s": 0
        },
        "target": "J_h",
        "unit": "STB/day/psi",
        "value": 7.5062338989974755
      },
      {
        "expression": "(0.00708 * k_h * h) / (mu_o * B_o * D_joshi)",
        "intermediates": {
          "A_ft2": 1742400,
          "r_eh": 744.7302502830383,
          "beta": 2,
          "a_joshi": 832.7873756978502,
          "R_h": 2.997542938886246,
          "D_joshi": 1.600469400319044
        },
        "inputs": {
          "k_h": 100,
          "k_v": 25,
          "h": 50,
          "L": 1000,
          "A_ac": 40,
          "r_w": 0.328,
          "mu_o": 2,
          "B_o": 1.2,
          "s": 0
        },
        "target": "J_h",
        "unit": "STB/day/psi",
        "value": 9.216046240596464
      },
      {
        "expression": "(0.00708 * k_h * h) / (mu_o * B_o * D_joshi)",
        "intermediates": {
          "A_ft2": 1742400,
          "r_eh": 744.7302502830383,
          "beta": 1,
          "a_joshi": 832.7873756978502,
          "R_h": 2.997542938886246,
          "D_joshi": 1.3144738074897397
        },
        "inputs": {
          "k_h": 100,
          "k_v": 100,
          "h": 50,
          "L": 1000,
          "A_ac": 40,
          "r_w": 0.328,
          "mu_o": 2,
          "B_o": 1.2,
          "s": 0
        },
        "target": "J_h",
        "unit": "STB/day/psi",
        "value": 11.221220168828001
      }
    ]
  }
}