In a classic paper, J. J. Arps^{[1]} took many observations from other investigators and concluded that the decline in the oil production rate, ${q}_{o}$, over time from actual oil reservoirs could be described by the equations:

where the decline rate, $D$, is a time dependent function:

Where:

- ${q}_{o}$ is the oil production rate, STB/day
- $t$ is the time, days (other units of time, such as months or years, can be used with an appropriate unit conversion constant)
- $D$ is the time dependent decline rate (time dependence defined by
**Equation 4.63**), days^{-1} - ${D}_{i}$ is the initial decline rate (constant), days
^{-1} - $b$ is a constant (typically used as a tuning parameter to match actual field data) and is in the range of $0\le b\le 1$, dimensionless

Decline curve analysis is essentially a curve fitting, or trend-line, analysis procedure where the form of the trend-line is developed from Arps^{[1]} observations (**Equation 4.68** and **Equation 4.69**). In this procedure, once the form of the trend-lines is established, we can use the parameters, ${q}_{oi}$, ${D}_{i}$,and $b$ to best match the data. We can develop these trend-lines or rate-time relationships if we integrate **Equation 4.68** with respect to time. The resulting relationships have three forms depending on the value of the b-parameter.

[1] Arps, J. J.: “Analysis of Decline Curves,” SPE-945228-G, Trans. of the AIME (1945)