Page 278 - Standard Handbook Petroleum Natural Gas Engineering VOLUME2

446 Reservoir Engineering

the production rate. For a harmonic decline, the decrease in production per
unit time as a fraction of the production rate is directly proportional to the
rate. Slider [256] presented an equation for the hyperbolic decline that will
reduce to the other types under certain circumstances:

(5-182)

where a is the decline rate when the production rate is q, and ai and q are
the decline rate and production at an initial time. As mentioned above, the
exponent, n, is a number between, but not including zero and one for the
hyperbolic decline. When n is zero, the decline rate is constant which is the
exponential decline. When n is one, the decline rate is proportional to the rate
which is the harmonic decline. Several early publications related to declinecurve
analysis have appeared in the literature [257-2621.
The exponential and hyperbolic types of decline curves are more common
than the harmonic decline. The exponential or constant percentage decline is
indicative of a homogeneous producing interval where the pressure response has
been affected by the outermost reservoir limits [263-2641. The exponential
decline may apply to pumping wells that are kept pumped off or gas wells and
many oil wells that produce at a constant bottomhole pressure. The hyperbolic
decline is indicative of either unsteady-state conditions or pressure response from
a variable permeability reservoir [265]. Although frequently encountered, the
harmonic decline may be observed with reservoirs that are dominated by gravity
drainage [197]. Equations for each type of decline-curve are given in Table 5-33,
and will be discussed for each case.

Exponential Decline
For the exponential or constant percentage decline, the nominal or instan-
taneous decline rate is:

a= qi /9) (5-183)
t
and as shown in Table 5-33, the rate-time relationship is:

q = qie" (5-1 84)

and the relationship between flow rate and cumulative production is:
91 -9
N, = - (5-185)
a
The annualized (effective) or continuous decline rate, d, is:

d = 91 -9 (5-186)
4
from which the cumulative production, N,, is:

91 -9
N, = (5187)
-tn(l- d)