Arps decline forecast
The rate after a given time and the cumulative production for exponential, hyperbolic or harmonic decline — from the initial rate, the first-year decline and the exponent b.
D_i = [(1 − D)^(−b) − 1] / b; b = 0: D_i = −ln(1 − D)q = q_i / (1 + b · D_i · t)^(1/b); b = 0: q = q_i · e^(−D_i · t)N_p = q_i^b · (q_i^(1−b) − q^(1−b)) / ((1 − b) · D_i); b = 0: (q_i − q) / D_i; b = 1: (q_i / D_i) · ln(q_i / q)D is the actual (effective) rate drop over the first year, as in "25 % per year"; the nominal decline D_i (1/yr) used in Arps' equations is found from it. With b = 0 the rate falls by the same fraction every year, with b > 0 the decline slows down over time. A year is 365 days. The equations are linear in the rate: enter t/d and the cumulative comes out in thousand tonnes. For a full well model — from the reservoir and the well type to a yearly production profile — see the Well production profile calculator.
Source: Arps J.J., Analysis of Decline Curves, Trans. AIME, vol. 160, 1945
Inputs
You can change a field's unit: the value is converted to the formula's units automatically.
Oil (or liquid) rate at the start of the forecast, on the stable decline trend.
The actual drop: 20 % means that after a year the rate is 0.8 · q_i.
0 is exponential, 1 harmonic, in between hyperbolic; fit it to the production history.
Unit converter for this formulaLiquid rate · Time · Liquid volume · Fraction and percent
- m³/d1
- m³/h0.0416667
- L/s0.0115741
- L/min0.694444
- m³/min0.000694444
- bbl/d6.28981
- Mbbl/d0.00628981
- ft³/d35.3147
- US gpm0.183453
- bbl/min0.00436792
A day is 24 hours.
- N_p — Cumulative production over t–
- D_i — Nominal initial decline rate–1/yr
More in Production forecast
A model of one oil well from start-up to the economic limit: inflow to a vertical, deviated, horizontal or hydraulically fractured well, transient and pseudo-steady flow, the bubble point, a liquid-rate plateau, water cut and a material balance with depletion or pressure support. Monthly rate and pressure charts, a yearly table and CSV export.
The time for the rate to fall to the economic limit and the production until then — the well's remaining recoverable reserves from the decline curve.
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.