This is the Arps decline-curve model — the standard reservoir-engineering tool used to forecast an oil well's production and estimate ultimate recovery (EUR) from historical rate data. The instantaneous rate follows:
q(t) = qᵢ / (1 + b·Dᵢ·t)^(1/b) 0 < b < 1 (hyperbolic)
q(t) = qᵢ · exp(−Dᵢ·t) b = 0 (exponential)
q(t) = qᵢ / (1 + Dᵢ·t) b = 1 (harmonic)
Cumulative production Np(t) is the closed-form time integral of q(t):
Np(t) = [qᵢ / ((1−b)·Dᵢ)] · [1 − (1+b·Dᵢ·t)^(1−1/b)] (hyperbolic)
Np(t) = (qᵢ/Dᵢ) · (1 − exp(−Dᵢ·t)) (exponential)
Np(t) = (qᵢ/Dᵢ) · ln(1 + Dᵢ·t) (harmonic)
The well is considered depleted once its rate falls to the economic limit qₗᵢₘ — the minimum rate that still covers operating cost. Solving q(t)=qₗᵢₘ for t gives the time-to-limit, and Np at that time is the Estimated Ultimate Recovery (EUR). b controls how quickly the decline rate itself decays: b=0 gives a constant fractional decline (straight line on the semi-log plot below), while b→1 gives a decline rate that itself keeps slowing, common in fractured/tight reservoirs with strong early-time heterogeneity.
- Main plot — q(t) on linear axes; the shaded area up to the time cursor is the cumulative volume produced so far.
- Cumulative pane — Np(t), asymptoting toward EUR as the well approaches its economic limit.
- Semi-log pane — log(q) vs t; exponential decline is a straight line here, hyperbolic and harmonic decline curve upward, which is exactly how engineers distinguish decline regimes from real field data.