Page 386 - Fundamentals of Reservoir Engineering

NATURAL WATER INFLUX 321

which on substituting in the inflow equation (9.18) gives

dW e p)e − Jp t / W ei (9.23)
i
i
dt = J(p −
Finally, integrating equ. (9.23) for the stated initial conditions yields the following
expression for the cumulative water influx

W − Jpt / W
W = ei (p − p)(1 e i ei ) (9.24)
−
e
p i i
What can be observed immediately from this expression is that as t tends to infinity,
then
W
W = p ei (p p)
e
i−
i
= cW (p − p)
i
i
which is the maximum amount of water influx that could occur once the pressure drop
p i - p has been transmitted throughout the aquifer.

As it stands, equ. (9.24) is not particularly useful since it was derived for a constant
inner boundary pressure. To use this solution in the practical case, in which the
boundary pressure is varying continuously as a function of time, it should again be
necessary to apply the superposition theorem. Fetkovitch has shown, however, that a
difference form of equ. (9.24) can be used which eliminates the need for superposition.
That is, for the influx during the first time step ∆t 1, equ. (9.24) can be expressed as

W − Jp t / W
∆
∆ W = ei p − p ) (1 e− i 1 ei ) (9.25)
1 e p i ( i 1
where p is the average reservoir boundary pressure during the first time interval. For
1
the second interval ∆t 2
W − Jp t / W
∆
∆ W = p i ei ( p − p 2 ) (1 e− i 2 ei ) (9.26)
e
1 a
2
where p is the average aquifer pressure at the end of the first time interval and is
1 a
evaluated using equ. (9.20) as

∆ W
i
p = p 1− i e (9.27)
1 a W
i e
th
In general for the n time period,
W − Jp t / W
∆
−
∆ W = p i ei ( p a n1 − p n ) (1 e i n ei ) (9.28)
e
n
−
where