Page 256 - Fundamentals of Reservoir Engineering

OILWELL TESTING 193

Since the production history of any oilwell consists of periods during which the rates
vary considerably, including periods of closure for repair and testing, it may be felt by
the reader that to interpret any buildup test conducted after a lengthy period of
production would require the application of the superposition principle as presented in
equ. (7.31) to obtain meaningful results.

Fortunately, this is not necessary providing that the well is producing under semi-
steady state conditions at the time of the survey. The following argument will show that,
in this case, the real time can be replaced by the effective flowing time, defined by
equ (7.14), without altering the value of the average pressure calculated from the
buildup analysis.

Suppose a well has been producing with a variable rate history prior to closure at real
time t n for a buildup survey. If the final production rate is q n during the period (t n − t n-1),
then the wellbore pressure at any time ∆t during the buildup can be determined using
the equation

kh n ∆ q j
7.08 10 -3 (p − p ) = p ( D t ) − p ( t ∆ ) (7.55)
×
t + ∆
qB o i ws j1 q n D n j D 1 D D
µ
−
n
=
which is simply a direct application of equ. (7.31) for the variable rate history, including
the buildup. It is analogous to the theoretical buildup equation, (7.32), which was
derived for constant rate production during the entire history of the well. Therefore,
repeating the steps taken in the derivation of equ. (7.37) from equ. (7.32), equ. (7.55)
can be expressed as

kh t +∆ t
7.08 10 -3 ( i p ) = 1.151 log n
×
p −
qB o ws(LIN) t ∆
µ
n
n ∆ q 4t (7.56)
t
+ j p D ( D + t ∆ ) − ½ ln D n
j1 q n n j D 1 γ
−
=
which is the theoretical linear equation which matches the actual buildup for small
values of ∆t. Implicit in the derivation of equ. (7.56) is the condition that the final flow
period (t n − t n- 1) >> ∆t, thus the last two terms in the equation are constants evaluated
at time t n.
Alternatively, if the effective flowing time t = N p/q is used in the analysis then a different
buildup plot will be obtained for which the early linear trend can be matched by
equ (7.48), in which the final flow rate is q n, i.e.

kh t +∆ t 4t
7.08 10 -3 (p − p ) 1.151 log + p () ½ ln D (7.48)
t
×
−
=
qB o i ws(LIN) t ∆ D D γ
µ
The two buildup plots for real and effective flowing time are shown as lines A and B,
respectively, in fig. 7.20.