4.5.2.1.1: Exponential Decline (b=0)

If $b = 0$ , then from Equation 4.69, $D = D_{i} = c o n s t a n t$ , and we can integrate Equation 4.68 to obtain:

$l o g_{e} (\frac{q_{o}}{q_{o i}}) = - D t$

Equation 4.70a

or,

$q_{o} = q_{o i} e^{- D t}$

Equation 4.70b

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 $(q_{o} - N_{p})$ relationship by noting that:

$N_{p} = \int_{0}^{t} q_{o} d t$

Equation 4.71

Multiplying Equation 4.70b by $d t$ and integrating results in:

$N_{p} = - \frac{(q_{o i} e^{- D t} - q_{o i})}{D} = \frac{q_{o i}}{D} (1 - e^{- D t})$

Equation 4.72a

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

$N_{p} = - \frac{(q_{o} - q_{o i})}{D} = \frac{(q_{o i} - q_{o})}{D}$

Equation 4.72b

or, finally:

$q_{o} = q_{o i} - D N_{p}$

Equation 4.72c

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, $q_{o a b}$ , (rate at which the revenue from the oil sales would pay for the operating expenses of the reservoir or well), then we would have:

$N_{p} = \frac{(q_{o i} - q_{o a b})}{D}$

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:

$N_{p} = \frac{(q_{o i} - 0)}{D} = \frac{q_{o i}}{D}$

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, $q_{o i}$ 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)

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