Page 388 - Fundamentals of Reservoir Engineering

NATURAL WATER INFLUX 323

dW − Jp t / W
q = e = J(p − p)e i ei (9.23)
w
i
dt
the steady state case implies that W ei, the encroachable water, is infinite and therefore

dW
q = e = J(p − p) (9.32)
w
i
dt
which, upon integration, gives the cumulative water influx as

t

W = J (p − p)dt (9.33)
e
i
0
Equs. (9.32) and (9.33), which are a special case of Fetkovitch's theory, were first
6
presented in 1936 by Schilthuis and described as steady state influx equations.
Equation (9.33) can be evaluated in stepwise fashion in which values of p , the inner
n
th
boundary pressure during the n time period, are calculated using equ. (9.15).
The reader should also be aware that the PI expressions presented in table 9.8 were
2
derived in similar form in Chapter 6, sec. 2, under the assumption that (r w/r e) was
2
approximately zero. For small radial aquifers, the equivalent assumption that (r o/r e) is
negligible may not always be applicable and the correct PI expression should then be
obtained by solving the radial diffusivity equation, using exactly the same steps as
shown in Chapter 6 but, without neglecting such terms. Considering the inherent
uncertainties in aquifer fitting this approach is generally unnecessary and, in fact,
Fetkovitch has demonstrated an almost perfect match between his results and those of
Hurst and van Everdingen for values of r eD as small as three.

For the case of a reservoir asymmetrically situated within a non-circular shaped aquifer
it should, with tolerable accuracy, be possible to use the Dietz shape factors presented
in fig. 6.4, and described in Chapter 7, sec. 7, to modify the semi-steady state PI
expressions. Thus the radial PI in equ. (9.30) can be generalised as

2fkh
π
J =
µ ln 4A
2 γ C r 2
A o
which has precisely the same form as equ. (6.22).

For very large aquifers, the initial flow of water into the reservoir will be governed by
transient flow conditions. In this case, it takes a finite time for the initial pressure
disturbance at the reservoir-aquifer boundary to feel the effect of the outer boundary of
the aquifer. Unfortunately, during this transient flow period it is no longer possible, in
analogy with wellbore equations, to derive a simple expression for the productivity
index J. This is because for inflow into a reservoir it is incorrect to use the approximate
line source solution of the radial diffusivity equation, in an attempt to evaluate a PI
under transient conditions, since r o is always finite and the boundary conditions for this
type of solution, expressed in equ. (7.1), can no longer be justified. Therefore, the
method of Fetkovitch cannot be used for the description of influx from an infinite aquifer