Environmental Engineering Reference
In-Depth Information
2.2 Deformable Porous Media
If the porous medium is deformable, additional equations are needed for a closed
system. Here we shall assume that the porous medium is heterogeneous and com-
posed of elastic material. A mesoscopic model of deformation of a porous elastic
solid is provided by the theory of elasticity modified to account for micro-structural
disorder. In this way, the heterogeneity of the solid matrix can be characterized by
the spatial variations of the local elastic constants. The mechanics of deformation
of heterogeneous media is largely based on studies of wave propagation (Karal and
Keller
1964
;Frisch
1968
;Weaver
1990
; Ryzhik et al.
1996
; Larose
2006
), and it is
in this context that we shall derive the general formalism to be implemented here.
The analysis of deformation of a heterogeneous solid body can be handled mathe-
matically by introducing the concept of a continuummedium. In this idealization, we
assume that the properties of the medium averaged over a mesoscopic scale are con-
tinuous functions of position and time. However, the presence of heterogeneities at
the microscopic scale, implying preferred force paths within the medium, have been
used as an empirical argument against an isotropic continuum description of inhomo-
geneous materials. Nonetheless, recent findings on the stress distribution response
to local and global perturbations have shed some light on the validity of a continuum
theory (Ellenbroek et al.
2009
).
For multiphase flow through a deformable porous medium, a new dependent vari-
able, say
w
R
(
, must be introduced for the displacement field of the solid matrix.
Furthermore, we shall assume that the deformations are small. While this assump-
tion limits the range of applicability of the theory, it is physically reasonable for
poroelastic materials under strong static compression like underground soil rocks.
We note that for infinitesimal deformations both the displacements and their gradi-
ents are much smaller than unity. With this provision and using index notation, the
components of the strain tensor of the solid can be defined up to linear order as
x
,
t
)
∂
1
2
w
R
,
i
∂
+
∂
w
R
,
j
∂
ʵ
ij
=
,
(35)
x
j
x
i
where
ʵ
ij
=
ʵ
ji
,
w
R
={
w
R
,
i
}=
(
w
R
,
1
,
w
R
,
2
,
w
R
,
3
)
=
(
w
R
,
x
,
w
R
,
y
,
w
R
,
z
)
, and
x
in Cartesian coordinates. Strain and stress are
linked by a stress-strain or constitutive relationship. The most general relationship
between the stress and strain tensors can be written as
={
x
i
}=
(
x
1
,
x
2
,
x
3
)
=
(
x
,
y
,
z
)
ʣ
ij
=
C
ijkl
ʵ
kl
,
(36)
where
C
ijkl
is a fourth-rank stiffness tensor having 81 components and
ʣ
ij
=
ʣ
ji
.
Because of the symmetry of the stress and strain tensors,
C
ijkl
has only 21 inde-
pendent components, which are necessary to specify the stress-strain relationship
for the most general form of the elastic solid. If the properties of such a solid vary
with direction, the material is termed anisotropic. In contrast, the properties of an
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