Page 280 - Fundamentals of Reservoir Engineering

OILWELL TESTING 217

Start of multi-rate test

q N
Rate

Time t N-1 t N t n

Fig. 7.29 Multi-rate test conducted after a variable rate production history

kh N
t
7.08 10 − 3 ( i p wf ) = ∆ q p D ( D + t δ D − t D ) + q S
p −
×
j
N
µ B o n j1 N n j 1 (7.74)
−
=
n
(
+ ∆ qp δ t D n − t δ D − j 1 ) (q − q N )S
+
D
j
n
jN 1
=+
in which t δ n = t − t N
n
and t δ j1 = t j1 − t for j N 1
≥
+
N
−
−
Then if the condition is imposed that (t N−t N-1)>> t n(max), i.e. the last flow period before the
test commences is considerably greater than the total duration of the test itself, then
N N
t
t
∆ qp D ( D N − t δ D n − t D j 1 ) ∆ q p D ( D N − t D j 1 ) (7.75)
≈
j
j
j1 − j1 −
=
=
and
N kh
t
∆ q p D ( D N − t D j 1 ) + q S ≈ 7.08 10 − 3 ( i p wf N )
×
p −
N
j
j1 − µ B o
=
where p wf N is the flowing pressure recorded immediately before the multi-rate test
commences.
Equation (7.74) can therefore be simplified as
kh N n
(
7.08 10 − 3 (p wf − p wf ) = ∆ q p δ t D − t δ D ) (q − q n )S
×
+
j
D
n
µ B o j1 N n j N 1 n j 1
−
= +
=
and therefore a plot of
(p wf N − p wf n ) versus n ∆ q j p δ t − t δ ) (7.76)
(

j 1
q − q N jN 1 q − q N D D n D −
=+
n
n
should again be linear with slope m = 141.2 µB o/kh and the intercept on the ordinate
equal to mS. Using this modified technique provides a useful way of applying the multi-
rate flow test for routine well surveys. The only condition for its application is that the
flow period before the test should be much longer than the totai test duration. This