Environmental Engineering Reference
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are constant values representing an equivalent averaged effective medium. The
average expectation values with respect to the corresponding probability distribu-
tion are denoted by
is a measure of the departure of the medium from
homogeneity. The fluctuations in relations (
38
) are assumed to be independent and
identically distributed functions with zero mean values
·
.Here
ʵ
)
=
ʻ
1
(
ˁ
R
,
1
(
x
x
)
=
μ
1
(
x
)
=
0
.
(39)
In addition, we assume that the fluctuations are homogeneous and isotropic with
respect to
x
. Thus, there would be nine autocorrelation functions defined as
x
)
=
ˁ
R
,
1
(
x
)
,
x
)
=
ˁ
R
,
1
(
)ʻ
1
(
x
)
,
R
(
x
|
x
)ˁ
R
,
1
(
R
1
(
x
|
x
ˁ
ˁ
1
ʻ
R
,
1
R
,
1
ˁ
R
,
x
)
=
ˁ
R
,
1
(
x
)
;
R
ˁ
R
,
1
μ
1
(
x
|
x
) μ
1
(
x
)
=
ʻ
1
(
x
)
,
x
)
=
ʻ
1
(
)ʻ
1
(
x
)
,
R
ʻ
1
ˁ
R
,
1
(
x
|
x
)ˁ
R
,
1
(
R
ʻ
1
ʻ
1
(
x
|
x
x
)
=
ʻ
1
(
x
)
;
R
ʻ
1
μ
1
(
x
|
x
) μ
1
(
x
)
=
μ
1
(
x
)
,
x
)
=
μ
1
(
)ʻ
1
(
x
)
,
R
ˁ
R
,
1
(
x
|
x
)ˁ
R
,
1
(
R
μ
1
ʻ
1
(
x
|
x
μ
1
x
)
=
μ
1
(
x
)
;
R
μ
1
μ
1
(
x
|
x
) μ
1
(
which are a measure of the spatial scale. For simplicity we shall assume that the
covariances are given by Gaussian random processes such that
x
)
=
x
|
),
R
ab
(
|
ʷ(
|
−
x
2
C
x
(40)
ˁ
R
,
ʻ
μ
where
a
and
b
stands for
.
We note that from the thickness of geological porosity logs (Hewett
1986
), it is
reasonable to expect a fractal character in the distribution of the micro-mechanical
properties of deformable porous media. Therefore, the concepts regarding frac-
tal geometry can be also applied for modelling the heterogeneity of oil reservoirs
(Srivastava and Sen
2009
). For instance, general forms for the self-affine fractal dis-
tributions of the bulk density, elastic moduli, and wave velocities of heterogeneous
materials can be modelled using the statistics of either a fractional Gaussian noise
(fGn) or a fractional Brownian motion (fBm) (Sahimi and Tajer
2005
). For example,
for a fBm,
, and
ʷ
in expression (
40
) is expressed as:
2
0
1
H
−
1
/
2
H
−
1
/
ʷ(
x
,ˉ)
=
ʷ
0
+
dB
(
s
,ˉ)
(
x
−
s
)
−
(
−
s
)
ʓ(
H
+
1
/
2
)
−∞
2
x
H
−
1
/
+
dB
(
s
,ˉ)(
x
−
s
)
,
(41)
0
for
x
>
0, where
ʷ
0
=
ʷ(
0
,ˉ)
,
H
is the Hurt exponent,
B
(
x
,ˉ)
is the Bachelier-
Wiener-Lévy process, and
ʓ
is the gamma function. For
H
>
1
/
2
(<
1
/
2
)
the spatial
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