8 CONCERNING THE APPARENT PERMEABILITY GIVEN BY WELL TESTS
We will now take a look at a more classical analysis of well tests. In the homogeneous case, it is recognized that well
tests are interpreted according to the following formula:
P r ¼ 0
, t
¼ Q
4pD fmc
t
3 t 32
Let us now assume that the reservoir is of infinite extent, heterogeneous and of stationary and isotropic heterogene-
ities. Our present goal is to show that for large times, we have the following behaviour:
P r ¼ 0
, t
¼ Q
4pD
eff
fc
t
3 t 33
where the equivalent diffusivity D
eff
is as defined in Sections 4, and 6. It is thus straightforward to procure an
effective permeability through the relation k
eff
¼ fmc
t
.D
eff
Here we will provide an heuristic proof. To begin with, the basic idea is to study the behaviour of the ensemble
average pressure hPr ¼ 0, ti. The difference with the pre- viously given formulation is the averaging symbol: well
tests are performed in one realization so the aim is to con- sider that the pressure diffusion takes into account the full
set of heterogeneities in a single realization for sufficiently long times. This regime is likely to occur when the investi-
gation radius—defined using the moment approach—is larger than the correlation length.
It would be thus sufficient to invert eqn 24 in r ¼ 0 to
obtain the expression of hPr ¼ 0, ti, but the 1t singularity gives rise to a diverging integral in the Fourier inversion. To
circumvent it, we compute the Fourier–Laplace transform of thPr ¼ 0, ti which is given by ¹ ] P
r ¼
, s
h i=]
s. As we are seeking the long-time behaviour of this function and as
we expect a constant, it is most logical to examine the quantity
lim
s→0
s 3 ]P
r ¼
, s
]s ¼
4pD
well test ¹
1
¼ D
eff ¹
1
34 Using the inverse Fourier transform, we must try to
evaluate s
]P r ¼
, s
]s ¼
s 4p
2
Z
dq
1 s þ D
q
2
þ q:
X
↔
q
, s
: q
2
þ
q ·
] X
↔
q
, s
]s
·q
s þ D q
2
þ q:
X
↔
q
, s
: q
2
2 6
6 6
6 6
6 6
4 3
7 7
7 7
7 7
7 5
35 It would be appealing to replace S
↔
q, s by S
↔
q ¼ 0, s ¼ 0,
and to compute the desired limit directly, but the integrals are not absolutely convergent and somewhat more calcula-
tion is needed. Rescaling the wave vector q by q ¼ q
s=
D p
q9 , we get a clearer form for the first term
of the preceding equation: s
4p
2
Z
d q
1 s þ D
q
2
þ q:
X
↔
q
, s
: q
2
¼ 1
4p
2
Z
d q
1 1 þ q
2
þ q:
X
↔
s
D r
q
, s
D :
q
2
36
Now, we can safely consider the limit s ¼ 0 under the integral sign, as the integral is absolutely convergent for
large q wave vectors. We thus obtain
D lim
s→0
1 4p
2
Z
dq
1 1 þ q
2
þ
q:
X
↔
s
D r
q
, s
D :
q
B B
B 1
C C
C A
2
B B
B B
B B
B B
B B
B B
B 1
C C
C C
C C
C C
C C
C C
C A
¼ 1
4p
2
Z
dq
1 1 þ q
2
þ q:
X
↔
, D
: q
B B
B 1
C C
C A
2
37
As S
↔
q ¼ 0, s ¼ 0 is isotropic, after integration, we pro-
duce the desired result:
D
well test
¼ D
1 þ
X
↔
q ¼ 0
, s ¼ 0
38 Now, we must show that the limit of the second term of the
right-hand side I
2
of eqn 35 is equal to zero in the low s limit. To achieve this task, we will explicitly use the
second-order expansion of the S
↔
q, s kernel whose expres-
sion in the real space is S
↔
r9
, t9
¼ j
2
==P
r9
, t9
C r9
, and its FLT is given by
X
↔
q
, s
¼ ¹ j
2
Z
dq9q9q9
1 s þ D
q9
2
C q ¹ q9
39 Using this result, along with the explicit form of the second
term, we get the following expression, after a rescaling of the wave vectors q and q9 by the factor
s=D
p :
588 B. Nœtinger, Y. Gautier
I
2
¼ s
4pD j
2
Z
dq
Z
dq9 qq : q9q9
3 1
1 þ q
2
þ q:
X
↔
s=D
p
q
, s
D :
q
B B
B 1
C C
C A
2
3 C
s=D p
q ¹ q9
1 1 þ q9
2 2
40
To examine the behaviour of this quantity, we first let the Laplace parameter s go to zero in the integrand. However,
this leads to the evaluation of two logarithmically divergent integrals. To obtain well-defined results, we return to eqn
34 and change the integration variable q9 by writing q9 ¼ q þ q0
. The integration over q is thus feasible and after tedious calculation, we are led to evaluate the limit for
small s of the following quantity: I
2
¼ s
Z
`
dq f q
C
s=D p
q in which the function fq is regular near zero, and behaves
as 1q for large q.
Assuming that the correlation function Cr has a limited
range l
c
in real space, we can restrict the q integration over
wave-vectors of modulus such that q ,
D =
s p
= l
c
. With this natural cut-off, it may be shown that the integral behaves as
s lns when s goes to zero. This means that, up to this order, the well test determined
permeability coincides with the stationary value for a suf- ficiently long time test. It must be stressed that the limit is
reached quite slowly, and that the well test interpretation method, in practice, could yield an apparent value quite
different from the theoretical one.
9 DISCUSSION AND CONCLUSIONS
Using a systematic perturbation expansion and Feynman graphs, we derived an exact integro-differential equation
which drives the average pressure in stationary hetero- geneous reservoirs. This equation is characterized by a
memory kernel S
↔
r, t which contains all the information
