Page 280 - Standard Handbook Petroleum Natural Gas Engineering VOLUME2

248 Reservoir Engineering

For the hyperbolic decline, the rate-cumulative production relationship is:

(5191)

and if h is substituted into this equation:
N,= [ - $)[ qi -’(’ +?)I (5192)

l)](
(h!

Equation 5-190 can be rearranged to:

l
l
-=-+- t
a ai h (5-193)
which represents a straight line. If the decline rate, l/a, is plotted on the y-axis
versus the time interval on the x-axis, the intercept at t = 0 will yield l/%, and
the slope will yield l/h. These values can be substituted into Equation 5-190 to
give any future estimates of production rates [264].
Harmonic Decline

For a harmonic decline, the time-rate relationship is:

(51 94)

and the rate-cumulative production relationship as shown in Table 5-33 is:
:)
N, = tn( (5-195)
ai

Production Type-Curves

Semilog Plots
The complexity of the analysis of hyperbolic decline-curves led to the develop
ment of curve-matching techniques. One of the simpler techniques was proposed
by Slider [256] with the development of an overlay method to analyze rate-time
data. The actual decline-curve data are plotted on transparency paper and
compared to a series of semilog plots that represent different combinations of
a, and n. Tabular values needed to plot the hyperbolic type-curves are available
[197] for values of n from 0.1 to 0.9, in increments of 0.1.
Gentry [266] prepared a series of plots of q/q versus NJqt for different
values of n from 0 to 1.0 in increments of 0.1. Using two rates, the cumulative
production, and the intervening time, the value of n for a particular hyperbolic