Arps Harmonic Decline Curve (Well Rate over Time)

The harmonic curve is the slowest of Arps' three decline curves. It gives the rate $q$ of an oil or gas well at time $t$ after the decline starts, from the initial rate $q_i$ and the initial nominal decline rate $D_i$, which is the fraction of the rate lost per year at the start. The rate falls as the reciprocal of one plus $D_i\,t$, so the decline rate itself falls as the well ages, and at long times the curve is flatter than the exponential one. It is the $b = 1$ case of the hyperbolic curve, and the exponential is the limit as $b$ goes to 0.

Only the product $D_i\,t$ matters, so time and decline rate must be in the same unit: a rate of 1.44 per year goes with years and 0.12 per month with months. At $D_i\,t = 1$ the rate has halved, and at $D_i\,t = 9$ it is a tenth of the start. The worked example is a well at 1 000 barrels a day with $D_i = 0.5$ per year. After 2 years $D_i\,t = 1$, so the rate is 500. After 6 years $D_i\,t = 3$ and the rate is a quarter of the start, 250. Run backwards, the same figures give $D_i = (1\,000/250 - 1)/6 = 0.5$ per year, and the time to fall to 100 barrels a day is $t = (1\,000/100 - 1)/0.5 = 18$ years.

The risk is in the tail. The area under the curve grows without bound as the logarithm of time, so the total volume the harmonic curve forecasts has no limit, and a forecast has to be stopped at an economic limit. Early data often cannot tell the harmonic curve from the exponential one, and the choice matters: in the example Fetkovich worked in 1980 from Arps' own data, a limit of 20 barrels a month gave a life of 285 months on the exponential curve and 1 480 months on the harmonic. The equation is Arps' rate-time relation as Fetkovich prints it, $q/q_i = 1/(1 + b\,D_i\,t)^{1/b}$, which reduces to $1/(1 + D_i\,t)$ for $b = 1$. The sources are Arps (1945), Analysis of decline curves, in the Transactions of the AIME, and Fetkovich (1980), in the Journal of Petroleum Technology.