If $b=0$, then from **Equation 4.69**, $D={D}_{i}=constant$, and we can integrate **Equation 4.68** to obtain:

or,

**Equation 4.70b** is one of the rate-time relationships observed by Arps^{[1]}. This equation is referred to as * Exponential Decline* because of the presence of the exponential term. We can also develop a rate cumulative production $\left({q}_{o}-{N}_{p}\right)$ relationship by noting that:

Multiplying **Equation 4.70b** by $dt$ and integrating results in:

or, after substituting **Equation 4.70b** into **Equation 4.72a**:

or, finally:

**Equation 4.72** is the rate cumulative production relationship for exponential decline. The form of **Equation 4.72b** has two important applications. First, if we know the * Abandonment Rate* for the reservoir or well, ${q}_{o\text{}ab}$, (rate at which the revenue from the oil sales would pay for the operating expenses of the reservoir or well), then we would have:

$${N}_{p}=\frac{\left({q}_{oi}-{q}_{o\text{}ab}\right)}{D}$$

This would tell us the volume of oil that the reservoir or well would produce above the economic threshold. The second application of **Equation 4.72b** is if we would like to determine the production from the reservoir or well at an infinite time regardless of the economics. The volume of oil that can be recovered from a reservoir or well with no regard to the economics is called the Estimated Ultimate Recovery, or EUR, of the reservoir or well. We can determine the EUR by simply allowing the rate from reservoir or well to decline to 0 STB/day production rate (infinite time). That is:

$${N}_{p}=\frac{\left({q}_{oi}-0\right)}{D}=\frac{{q}_{oi}}{D}$$

The form of **Equation 4.72c** has one important application: to make future well forecasts. We can see that **Equation 4.72c** is a straight line in ${N}_{p}$ with a slope of $-D$ (compare this straight-line relationship to the plot in **Figure 4.09**). Exponential decline is most often associated with the Rock and Fluid Expansion Drive Mechanism. In exponential decline, we have two parameters, ${q}_{oi}$ *and* $D$, with which to match the field data.

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