Arps Hyperbolic Decline Curve (Well Rate over Time)

Every oil or gas well gives less as it ages, and this formula says how quickly, in the form J. J. Arps published in 1945. The well starts on decline at a rate $q_i$ and, at that moment, loses a fraction $D_i$ of its rate per unit time: the initial nominal decline rate. The exponent $b$ says how fast that decline rate itself slows as the well ages. At $b = 0$ it never slows and the curve is exponential. At $b = 1$ it slows as fast as it can and the curve is harmonic. Between the two, the classical range of $b$ from 0 to 1, the curve is the hyperbola this page solves.

The exponential and harmonic curves have pages of their own, because at $b = 0$ the exponent $1/b$ cannot be formed, so the exponential is a limit of this formula, and at $b = 1$ the formula gives exactly the simpler harmonic curve. Time and the decline rate must be in the same unit, so 0.12 per month goes with months and 1.44 per year with years. Take a well that starts at 1 000 barrels a day with $D_i = 0.5$ per year and $b = 0.5$. After two years the bracket $1 + b\,D_i\,t$ is $1 + 0.5 \times 0.5 \times 2$, and raising it to the power $1/b = 2$ gives 2.25. The rate is 1 000 divided by 2.25, which is 444.444 barrels a day. Run backwards, the same 444.444 after two years means an initial rate of 1 000, and at the same $D_i$ and $b$ the time to fall to 100 barrels a day is 8.649111 years.

The real numbers come from Fetkovich's 1980 type-curve match of Arps' own hyperbolic example: an initial rate of 30 303 barrels a month, $D_i$ of 0.12 per month and $b = 0.5$. At 100 months the curve gives 30 303 divided by 49, which is 618.43 barrels a month. The same paper shows how much the choice of curve matters: at a limit of 20 barrels a month, the exponential curve gives a life of 285 months, against 1 480 months for the harmonic.

A fitted $b$ above 1 turns up for wells whose early flow is still transient, such as wells in very tight rock. The page allows it: with $b = 1.5$ and the same 1 000 barrels a day, 0.5 per year and two years, the rate is 542.88 barrels a day. The caution is that such a curve gives an unbounded total volume, so it must be ended by an economic limit or by a switch to exponential decline.