Page 375 - Fundamentals of Reservoir Engineering

NATURAL WATER INFLUX 310

(p + p ) p − p 1
i
1
i
∆ p = p − p 1 = p =
0
i
i−
2 2
(p + p ) (p + p ) p − p 2
2
1
1
i
i
∆ p = p − p 2 = − =
1
1
2 2 2
(p + p ) (p + p ) p − p
∆ p = p − p 3 = 1 2 − 2 3 = 1 3
2
2
2 2 2
(9.16)

(p + p ) (p + p ) p − p j 1
j 1
j 1
j 1
j
j
∆ p = p − p j 1 = − − + = − +
j
j
+
2 2 2
Therefore, to calculate the cumulative water influx W e at some arbitrary time T, which
th
corresponds to the end of the n time step, requires the superposition of solutions of
type, equ. (9.5), to give
W e (T) = U [∆p oW D (T D)+∆p 1 W D (T D - t D1)+ ∆p 2 W D (T D - t D )
2
+………….∆p jW D (T D - t D j)+…∆p n-1 W D (T D - t D n-1)]
where ∆p j is the pressure drop at time t j, given by equ. (9.16), and W D (T D - t D ) is the
j
dimensionless cumulative water influx, obtained from figs. 9.3 - 9.7, for the
dimensionless time T D - t D during which the effect of the pressure drop is felt. Summing
j
the terms in the latter equation gives
n1
−
WT ∆ p W (T − T ) (9.17)
() U=
D
j
D
e
j D
j0
=
In the special case of an infinite, linear aquifer for which, as noted in sec. 9.2, there is
no W D function included in figs. 9.3 - 9.7, the cumulative water influx at time T due to a
step-like pressure decline at the aquifer-reservoir boundary can be calculated using
equ. (9.13) as
−
φ kc n1
WT 2hw ∆ p j T − t j
() =
e
πµ j0
=
-3
which, when expressed in field units has the same constant, 3.26×10 , as equ. (9.14).
The following exercise will illustrate the application of the method of superposition in
history matching.
EXERCISE 9.2 AQUIFER FITTING USING THE UNSTEADY STATE THEORY OF
HURST AND VAN EVERDINGEN
A wedge shaped reservoir is suspected of having a fairly strong natural water drive.
The geometry of the reservoir-aquifer system is shown in fig. 9.10.