Geoscience Reference
In-Depth Information
The fluid displacement terms in Darcy's law can be eliminated by multiplying each of
(8.74) first by
ρ
w
and
ρ
a
, respectively, and then taking the divergence, after which substi-
tution of the continuity equations (8.61) and (8.62) gives
k
w
ρ
w
μ
w
∇
p
w
∂
∂
t
ρ
w
n
0
S
∂
u
∂
t
[
ρ
w
n
0
S
]
+∇ ·
=∇ ·
(8.76)
k
a
ρ
a
μ
a
∇
p
a
∂
∂
t
ρ
a
n
0
(1
−
S
)
∂
u
∂
t
[
ρ
a
n
0
(1
−
S
)]
+∇ ·
=∇ ·
Equations (8.75) and (8.76), together with solid continuity (8.63), form a closed system.
Hence it should be possible to solve it for any consolidation or flow problem involving an
elastic porous material occupied by two elastic fluids. Fortunately, for many problems this
formulation is more general than necessary (see, however, Verruijt (1969)), and it is often
possible to simplify it considerably as follows.
Simple case of constant vertical load
In soil mechanics and groundwater hydraulics the problem formulation is commonly sim-
plified by two basic assumptions that were introduced by Terzaghi (1925, 1943) and Jacob
(1940, 1950). First, compression is assumed to be strictly vertical without any horizontal
solid displacements, and second, any changes in vertical compressive effective stress are
balanced by equal and opposite changes in fluid stresses. These assumptions may be diffi-
cult to justify, but the resulting formulation has been used quite successfully in the solution
of many problems in porous media saturated with one fluid. This suggests that the concept
may also be valid in the simplification of certain problems involving two immiscible fluids.
In the present notation the first assumption can be written as
u
x
=
u
y
=
0, so that
e
zz
=
e
and the third of Equations (8.67) or (8.70) (for incompressible grains) yields
τ
zz
=
(2
μ
+
λ
)
e
(8.77)
The second assumption can be written as
τ
zz
=−
(
τ
w
+
τ
a
), which yields with (8.64) and
(8.65)
τ
zz
=
χ
p
w
+
(1
−
χ
)
p
a
(8.78)
Note as an aside, that Equation (8.78) agrees with the general observation that soils that are
close to saturation do not easily disintegrate, but exhibit a certain degree of consistency and
coherence. Indeed, if the effect of the air pressure can be neglected, in a soil that is close
to saturation, the water pressure
p
w
is negative and the effective stress factor
χ
is close to
unity; hence the intergranular stress
τ
zz
is also negative, which means that the soil grains
are drawn together and the soil exhibits a greater firmness. This effect can also be seen, for
example, on a sandy beach just after the sea water has withdrawn during ebb tide; at that
time the sand forms a harder surface than when it is submerged, with
p
w
>
0, or than when
it is totally dry.
Combining Equations (8.77) and (8.78) one obtains, instead of (8.75), simply
(2
μ
+
λ
)
e
=
χ
p
w
+
(1
−
χ
)
p
a
(8.79)
Substitution of (8.63) into the first of (8.76) yields
k
w
ρ
w
μ
w
p
w
ρ
w
S
∂
e
ρ
w
β
w
∂
p
w
n
0
ρ
w
∂
S
∂
t
+
n
0
S
∂
t
+
∂
t
=∇ ·
∇
(8.80)
