Geoscience Reference
In-Depth Information
wetting fluid is forced into the porous material, while the total volume of the other two is
somehow kept constant, i.e. when
e
and
e
a
are kept zero. This means that owing to (8.67),
c
w
=
χ
K
w
/
n
0
S
(8.68)
where
K
w
=
β
−
1
w
is the bulk modulus of the wetting fluid and
β
w
its compressibility.
Similarly, for the non-wetting fluid
c
a
=
(1
−
χ
)
K
a
/
n
0
(1
−
S
)
(8.69)
where
K
a
is the bulk modulus of the non-wetting fluid. The coefficients
c
sw
,
c
sa
,
and
c
wa
indicate that, in principle at least, there should be a coupling between the volume changes
of the three constituents as indicated by the subscripts. However, as is now shown, these
coefficients are equal to zero if the density of the solid grains can be assumed to be constant.
To see this, eliminate the fluid strains from Equation (8.67) to obtain an alternative form of
the generalized Hooke law, namely,
τ
xx
=
2
μ
e
xx
+
(
λ
+
c
1
)
e
+
c
2
τ
w
+
c
3
τ
a
τ
yy
=
2
μ
e
yy
+
(
λ
+
c
1
)
e
+
c
2
τ
w
+
c
3
τ
a
(8.70)
τ
zz
=
2
μ
e
zz
+
(
λ
+
c
1
)
e
+
c
2
τ
w
+
c
3
τ
a
together with the fourth, fifth, and sixth of (8.67). The new constants are
c
1
=
2
c
sw
c
sa
c
wa
−
c
sw
c
a
−
c
sa
c
w
c
,
c
2
=
(
c
sw
c
a
−
c
sa
c
wa
)
/
c
,
c
3
=
(
c
sa
c
w
−
c
sw
c
wa
)
/
c
,
and
c
=
c
w
c
a
−
c
wa
Consider now a thought experiment in which
τ
xx
, the effective stress, is held constant
but the fluid pressures are increased. In the case of a porous material containing only one
fluid, this can be accomplished by placing an unjacketed saturated sample in the fluid and
then increasing the fluid pressure. In the case of material containing air and water, this can
be accomplished by increasing
p
w
and
p
a
in such a way that their difference
p
c
remains
constant. It is clear that if the solid grain density is constant, this process does not increase the
effective stresses, nor does it result in any solid displacement. Thus Equation (8.70) shows
that
c
2
=
0. In other words, if the solid material (not the solid frame) is incompressible,
one has also
c
sw
=
c
3
=
0 as well. Moreover, if this is
the case, one can relate
μ
and
λ
to the bulk modulus of the solid frame, as follows:
c
sa
=
0, which immediately results in
c
1
=
2
3
μ
+
λ
K
s
=
(8.71)
Stress versus rate of strain relationships for fluids
As noted in Section 8.3.1, Darcy's law represents the equation of creeping motion in porous
material. When the motion takes place within the pores while the porous material itself
is being subjected to deformation, the Darcy flux must be taken as the relative motion
between the solid matrix and the fluids. Apparently (Verruijt, 1969), this concept was first
proposed for liquid saturated media by Gersevanov around 1934. Biot did not use it in
his original paper (Biot, 1941), but he introduced it in his generalized theory of elastic-
ity (Biot, 1955). The displacement vector of the solid is an actual displacement length,
whereas the displacement vectors of the two fluids are defined herein as volumes when
multiplied by the total bulk cross-sectional area normal to them. Thus the relative velocity
