Horizontal well productivity (Joshi)
Steady-state productivity index of a horizontal well by Joshi's equation with permeability anisotropy, compared with a vertical well in the same drainage area.
J_h = 2π · k_h · h / (μ · B · (ln((a + √(a² − (L/2)²)) / (L/2)) + (I · h / L) · (ln(I · h / (r_w · (I + 1))) + s)))a = (L/2) · √(0.5 + √(0.25 + (2 · r_eh / L)⁴)), I = √(k_h / k_v)J_v = 2π · k_h · h / (μ · B · ln(r_eh / r_w))Steady-state single-phase inflow to a horizontal well midway through the pay, with a constant pressure at the drainage boundary. Valid for L/2 < 0.9 · r_eh and a well much longer than I · h. q = J_h · (P_e − P_wf).
Source: Joshi, JPT (1988); anisotropic form: Economides, Hill, Ehlig-Economides, Petroleum Production Systems (1994)
Inputs
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From core or tests; often k_v ≈ 0.1–0.3 · k_h. k_v = k_h for an isotropic reservoir.
Producing length of the wellbore in the pay zone.
Radius of the circle with the drainage area: √(A/π). L/2 must be below 0.9 · r_eh.
0.078 m for a 6⅛″ (155.6 mm) bit, 0.108 m for 8½″ (215.9 mm).
Referred to the well length: ΔP_s = q · B · μ · s / (2π · √(k_h · k_v) · L); 0 for an undamaged well.
Unit converter for this formulaPermeability · Length · Viscosity · Oil formation volume factor · Productivity index
- mD1
- D0.001
- µm²0.000986923
- 10⁻³ µm²0.986923
1 D = 0.9869 µm². Russian-language reports often give permeability in 10⁻³ µm², which is almost mD (1 mD = 0.9869 · 10⁻³ µm²).
- J_v — Vertical well in the same area (s = 0)–
- J_h/J_v — Ratio J_h / J_v–
- a — Half major axis of the drainage ellipse–
More in Inflow and flow in porous media
Liquid flow rate in linear flow by Darcy's law, in barrels per day.
Well flow rate for radial liquid flow by Darcy's law, in barrels per day.
Well flow rate for radial flow by Darcy's law, corrected by the oil formation volume factor, in cubic metres per day.
Theoretical productivity index of a vertical well in a bounded drainage area (circle, square, rectangle) from permeability, thickness, fluid properties and skin, and the same without skin. The drainage radius comes from the area per well.
Results are engineering estimates from standard formulas; for design decisions check them against the codes, project documents and specialists' calculations. The formulas carried over from the original set are unchanged, and their errors are described in the notes.