Geoscience Reference
In-Depth Information
affected by the effective stress path, the pore geometries, the wetting angle, and probably
other factors. The nature of
χ
has been the subject of uncertainty (see McMurdie and Day,
1960; Blight, 1967; Snyder and Miller, 1985). For the sake of simplicity, for small elastic
displacements it has been assumed earlier (Brutsaert, 1964) that
χ
=
S
. In general, it is
probably safe to assume that
χ
is a unique function of
S
, i.e.
χ
=
χ
(
S
). By applying
Bishop's proposal to the air phase, one can write for the part of the total stress acting on the
air
τ
a
=−
(1
−
χ
)
p
a
(8.65)
where
p
a
is the incremental air pressure over and above the initial pressure
p
ai
,
existing
prior to the displacement. Thus the total air pressure is
p
a
=
p
ai
+
p
a
.
When the two immiscible fluids are in a “funicular” state, that is, consisting of connected
filaments without isolated drops or occluded bubbles of entrapped gas, the total pressure
p
w
can be related to the total pressure
p
a
by the capillary pressure
p
c
, which is defined as
follows:
p
w
=
p
a
+
p
c
(8.66)
This capillary pressure equals the pressure decrease across the air-water interface considered
in the derivation of Laplace's equation (8.3) via (8.4). Hysteresis (see Section 8.2.3) is
always present. However, if the process in question involves only wetting, or only drying,
so that hysteresis effects are avoided,
p
c
=
p
c
(
S
,
n
0
) can be taken as a function only of the
saturation and of the porosity.
The components of the stress tensor and the changes of the displacements of the three
phases must satisfy the equations of motion (Brutsaert, 1964) or for slow displacements
the simpler equilibrium equations (Verruijt, 1969). However, these are not needed in the
present derivation.
Stress-strain relationship
If the solid strains and the changes in fluid content are small and if the processes involved
are reversible, the stress components may in general be assumed to be linear functions of the
strain components (Biot, 1941, 1955). This assumption yields a generalization of Hooke's
law (Brutsaert, 1964) in the case of an isotropic porous material:
τ
xx
=
2
μ
e
xx
+
λ
e
+
c
sw
e
w
+
c
sa
e
a
τ
yy
=
2
μ
e
yy
+
λ
e
+
c
sw
e
w
+
c
sa
e
a
τ
zz
=
2
μ
e
zz
+
λ
e
+
c
sw
e
w
+
c
sa
e
a
τ
xy
=
2
μ
e
xy
τ
xz
=
2
μ
e
xz
τ
yz
=
2
μ
e
yz
(8.67)
τ
w
=
c
sw
e
+
c
w
e
w
+
c
wa
e
a
τ
a
=
c
sa
e
+
c
wa
e
w
+
c
a
e
a
in which
μ, λ,
c
sw
,
c
sa
,
c
w
,
c
a
,
and
c
wa
are constants characterizing the elastic behavior of
the material. These equations reduce to Biot's (1955) when the pores are saturated with one
fluid and to Hooke's law for isotropic bodies if only the solid were present. Thus
μ
and
λ
represent the behavior of the solid. A fluid displacement does not result in a shear stress,
only the rate of displacement does; therefore the fluid strains do not appear in the shear
stress components. The coefficient
c
w
can be understood by considering the situation where
