Page 385 - Fundamentals of Reservoir Engineering

NATURAL WATER INFLUX 320

the same way as the flow of oil from a reservoir into a well. An inflow equation of the
form

dW
q = e = J(p − p) (9.18)
a
w
dt
is used where,

= water influx rate
q w
J = aquifer productivity index

p = reservoir pressure, i.e. pressure at the oil or gas water contact.

p a = average pressure in the aquifer.

The latter is evaluated using the simple aquifer material balance

W e=c W i (p i - p a) (9.19)

in which p i is the initial pressure in the aquifer and reservoir. This balance can be
alternatively expressed as

W W
i
i
p = p 1− e = p 1− e (9.20)

a
cWp W ei
ii
where W ei = c W ip i is defined as the initial amount of encroachable water and
represents the maximum possible expansion of the aquifer. Differentiating equ. (9.20)
with respect to time gives

dW e W ei dp a (9.21)
dt =− p i dt

and substituting equ. (9.21) into equ. (9.18) and separating the variables gives

dp Jp
a =− i dt
p − p W ei
a

This equation will now be integrated for the initial condition that at t=0 (W e = 0, p = p i)
a
a pressure drop ∆p = p i - p is imposed at the reservoir boundary. Furthermore, the
boundary pressure p remains constant during the period of interest so that

Jp t
ln ( p − ) p = − W ei i + C
a

where C is an arbitrary constant of integration which can be evaluated from the initial
conditions as C = In(p i - p), and therefore

i
pp−= ( p p e− ) − Jp t / W ei (9.22)
a
i