Page 232 - Fundamentals of Reservoir Engineering

OILWELL TESTING 170

2 π kh
p −
1
D
( i p wf ) = (q − 0) (p (t − 0) + S)
D
µ n n
+ (q − q ) (p (t D n − t ) + S)
2
D
D
1
1
+ (q − q ) (p (t D n − t ) + S)
D
2
3
D
2
.
.
+ (q − q ) (p (t − t D − ) S)
+
D
D
j
j 1
−
n
j 1
.
.
+ (q − q ) (p (t − t D − ) + S)
n 1
D
n
D
−
n 1
n
in which p wf n is the specific value of the bottom hole flowing pressure corresponding to
th
the total time t n which may occur at any time during the n period of constant flow,
when the rate is q n. In this summation all the skin factor terms disappear except for the
last, q n S. The summation can be expressed as
2 π kh n
t
( p − p wf ) = ∆ q p D ( D − t D ) + q S (7.31)
n
j
i
−
µ n j1 n j1
=
in which q∆ j = q − q j 1
j
−
Equation (7.31) may be regarded as the basic equation for interpreting the
pressure-time-rate data collected during any well test, and with minor modifications,
described in Chapter 8, can equally well be applied to gas well test analysis. The whole
philosophy of well testing is to mechanically design the test with a series of different
flow rates, some of which may be zero (well closed in), for different periods of time so
that equ. (7.31) can be readily interpreted to yield some or all of the required reservoir
parameters, p i, p, k, S, A and C A. The three most common forms of well testing are the
single rate drawdown test, the pressure buildup test and the multi-rate drawdown test.
The analysis of each of these tests using equ. (7.31) is briefly described below and in
much greater detail in the following sections of this chapter.
a) Single rate drawdown test
In this type of test the well is flowed at a single constant rate for an extended period of
time so that
q = q ; ∆ q = q and t D n = t D
1
1
and equ. (7.31) can be reduced to

2 kh
π
(p − p wf ) = p (t ) + S (7.19)
D
i
D
q µ
which is simply the constant terminal rate solution expressed in dimensionless form.
The flowing pressure p wf , which is recorded throughout the test, can be analysed as a