Page 364 - Fundamentals of Reservoir Engineering
P. 364
NATURAL WATER INFLUX 299
qµ
qt (9.3)
() =
D
D
2kh p
π
∆
where q D (t D) is the dimensionless influx rate evaluated at r D = 1 and describes the
change in rate from zero to q due to a pressure drop ∆p applied at the outer reservoir
boundary r o at time t = 0. These functions can be generated from constant terminal rate
solutions and vice-versa. It is generally more convenient to express this solution in
terms of cumulative water influx rather than rate of influx. Thus integrating equ. (9.3)
with respect to time
t D t
µ q (t ) dt dt
2kh p qdt = D D dt D D
π
∆
o o
which gives
W µ = Wt φµ cr 2 0
()
e
2kh p D D k
∆
π
and therefore
2
()
W e = 2πφ hcr ∆ pW t D (9.4)
D
0
in which, since Darcy units are being employed
= cumulative water influx (ccs) due to a pressure drop ∆p (atm)
W e
imposed at r o at t = 0
and W D (t D) = dimensionless, cumulative water influx function giving the
dimensionless influx per unit pressure drop imposed at the reservoir
aquifer boundary at t=0.
Equation (9.4) is frequently expressed as
W e = U p W (t ) (9.5)
∆
D
D
where
U = 2 f hcr o 2 (9.6)
πφ
which is the aquifer constant for radial geometry
and
(encroachment angle) °
f =
360 °
which is to be used for aquifers which subtend angles of less than 360° at the centre of
the reservoir-aquifer system.
The dimensionless water influx W D (t D) is frequently presented in tabular form or as a
set of polynomial expressions giving W D as a function of t D for a range of ratios of the