Geoscience Reference
In-Depth Information
system, Equations (3.8) are written as
phenomenological approach, strongly depend on
the microstructure. To quantify the structure-
sensitive behavior of
S
e
ijkl
and
S
v
ijkl
, various
theoretical models assuming various pore geome-
tries have been developed. These models can be
roughly classified as either ''inclusion models''
or ''granular models.''
Inclusion models have been used to investigate
the elastic properties. In these models, the liq-
uid phase is modeled in terms of inclusions that
are embedded in the continuum of a solid phase.
The elastic compliance
S
e
ijkl
can be calculated as
a function of porosity and pore geometry, using
both physically and mathematically reasonable
procedures (e.g., O'Connell & Budiansky, 1974;
Berryman, 1980; Mavko, 1980). For mathematical
simplicity, the inclusions are simply assumed to
have the shape of, for example, an oblate spheroid,
but with a variable aspect ratio
δ
ij
f
ij
1
3
k
sk
1
(
p
S
p
L
)
S
ij
p
S
ε
=−
−
δ
ij
+
σ
+
2
µ
sk
3
k
S
p
L
1
−
δ
ij
(elastic)
f
ij
1
1
S
ij
ξ
sk
(
p
S
p
L
)
p
S
ε
=−
−
δ
ij
+
η
sk
(
σ
δ
ij
)
(viscous),
+
3
2
(3.9)
where
p
S
S
kk
=−
σ
/
3 represents the solid pressure
(compression positive),
k
sk
and
µ
sk
the bulk and
shear moduli, respectively, and
η
sk
the
bulk and shear viscosities, respectively, of the
solid framework (under a drained condition).
In the next section, the elastic and viscous con-
stitutive relationships are derived theoretically in
the form of Equations (3.8) or (3.9). Equations (3.9)
can be modified to
ξ
sk
and
. Although the
significant effects of pore geometry are demon-
strated quantitatively, further application of these
models to experimental studies of pore geometry
prove to be difficult because of the oversimplified
shape of the inclusions. Inclusion models, assum-
ing the continuum of a solid phase, do not include
grain boundaries. Therefore, their application to
rheology proves to be difficult, too, because grain
boundaries play important roles in the anelastic
and viscous behavior of rocks as slip planes and
fast diffusion pathways.
Granular models have been used to investi-
gate viscosity and anelasticity, so that slip and/or
diffusion at grain boundaries can be treated explic-
itly (e.g., Raj, 1975; Cooper
et al
., 1989). However,
the granular models used to study rheology have
been too primitive to apply to elasticity. Takei
(1998a) developed a 3D granular model, in which
the elastic properties can be calculated for an ar-
bitrarily given grain-to-grain contact state. The
geometrical parameter in this model is given by
a contact function or ''contiguity'' (Figure 3.1c).
This contiguity model was first developed for
the elasticity of partially molten rocks (Takei,
1998a) and was subsequently extended to their
viscosity (Takei & Holtzman, 2009a). The appli-
cability of the contiguity model to anelasticity
was also demonstrated for melt-free aggregates
α
2
N
σ
ij
+
δ
ij
−
f
ij
1
3
K
b
1
1
3
k
S
p
L
p
L
)
ε
=−
(
p
−
δ
ij
+
p
δ
ij
(elastic)
f
ij
1
3
1
2
p
L
)
ε
=−
(
p
−
δ
ij
+
(
δ
ij
)
(viscous),
σ
ij
+
p
ξ
η
(3.10)
σ
ij
S
p
L
where
δ
ij
represents the av-
erage stress (tension positive) and
p
σ
ij
=
(1
−
φ
)
−
φ
=−
σ
κκ
/
3
=
p
L
the average pressure (compression
positive) of a two phase system. In numerical
simulations and experimental studies, the macro-
scopic stress state is represented by
)
p
S
(1
−
φ
+
φ
σ
ij
and
p
L
,
σ
ij
S
and
p
L
, and hence Equations
(3.10) are more commonly used than (3.9).
The bulk modulus
K
b
, shear modulus
N
,bulk
viscosity
rather than by
of the solid
framework, defined by Equations (3.10), are
simply related to those defined by Equations (3.9)
as
K
b
=
ξ
, and shear viscosity
η
(1
−
φ
)
k
sk
,
N
=
(1
−
φ
)
µ
sk
,
ξ
=
(1
−
φ
)
ξ
sk
,
η
=
−
φ
η
sk
.
and
(1
)
3.4 Contiguity Model
3.4.1 Comparisons with other models
The mechanical properties of a solid frame-
work (under a drained condition), namely
S
e
ijkl
and
S
v
ijkl
,
introduced above as part of
the
