Geoscience Reference
In-Depth Information
Once the time elapsed since injection started becomes larger than a fraction, say 0.1,
of the characteristic time
t
*
=R
2
/c
, then the evolution of the induced pore pressure becomes
influenced by the finiteness of the reservoir. Formally, the pore pressure solution can then
be expressed as
Δρ(
r,t
) = ρ
*
Ρ(
r/R, t/t
*
)
(2)
The function P(ρ,
t
) can be determined semianalytically. If the elapsed time
t
is expressed
as the ratio of the injected volume
V
to the rate of injection
Q
o
(i.e.,
t=V/Q
o
)
,
then solu-
tion (2) can be written as
Δρ(
r,V
) = ρ
*
Ρ(
r/R ,V/V
*
)
(3)
where
V
*
=(
Q
o
R
2
)/
c
is a characteristic fluid volume. The above expression suggests that the
relationship between the induced pore pressure Δρ, the injected volume
V
, and the injec-
tion rate
Q
o
is not straightforward. However, Equation (3) shows important trends; for
example, a decrease of the permeability causes an increase of the characteristic pressure,
or an increase of the storage coefficient causes a decrease of the pore pressure, all other
parameters kept constant.
At small time
t
t
*
, the dimensionless pressure P = Δρ/ρ
*
reduces to the unbounded
domain solution
F
, while at large time
t
≪
t
*
, the pressure tends to become uniform and the
≫
pore pressure is simply given by
ρ
V/(
π
R
2
HS)
(4)
≅
as the function P(ρ,t) behaves for large
t
as P
t/
π. Thus, at large time, the pore pressure
is simply proportional to the volume of injected fluid (Figure H.1). Equation (4) actually
indicates that the large-time pore pressure is simply the ratio of the injected volume over
the reservoir volume, divided by the storage coefficient.
The previous material provides some information about the link between pore pressure,
injected volume, and injected rate for the particular case of an injector centered in a disk-
shaped reservoir. These ideas can be generalized to more realistic cases. For example, for an
arbitrarily shaped reservoir with
n
wells, each injecting at a rate
Q
o
, the general expression
for the induced pore pressure can be written as
≅
Δρ(x,
t
) = ρ
*
ς{x/
L, t/t
*
; n,
(
x
i
, i =1, n
)
, reservoir shape}
where the characteristic pressure and time are given by
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