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Arps decline curves describe rate against time with a single exponent b: exponential (b = 0) for solution-gas drive and pressure depletion, harmonic (b = 1) for gravity drainage, and hyperbolic in between, which fits most unconventional wells. Cumulative production is the integral of the rate curve.
Exponential decline
q(t) = qi x e^(-D t); Np = (qi - q) / D
Hyperbolic decline
q(t) = qi x (1 + b D t)^(-1/b); Np = qi / (D (1-b)) x (1 - (q/qi)^(1-b))
Reserve estimates derived from decline curve analysis must be prepared or audited by a qualified reserves evaluator for any financial or regulatory reporting.
Fit it to at least 12-18 months of production history. Shale wells commonly show b between 0.8 and 1.4 early on, transitioning to exponential later, so a terminal decline rate must be imposed.
Harmonic and high-b hyperbolic curves predict infinite cumulative production. A terminal exponential decline, typically 5-8% a year, must be applied to keep reserve estimates physical.