Geoscience Reference
In-Depth Information
in which
β
w
[
=
(
∂ρ
w
/∂
t
)
/
(
ρ
w
∂
p
w
/∂
t
)] is assumed to be a constant, as a measure of the
compressibility of the water, and in which (
∂
u
/∂
t
)
·∇
(
ρ
w
S
) is assumed to be negligible on
account of the small solid velocity. Similarly, the second of (8.76) reduces to
k
a
ρ
a
μ
a
p
a
ρ
a
β
a
∂
p
a
S
)
∂
e
n
0
ρ
a
∂
S
ρ
a
(1
−
∂
t
+
n
0
(1
−
S
)
∂
t
−
∂
t
=∇ ·
∇
(8.81)
in which now (
S
)] is assumed negligible. The combination of (8.79)
with (8.80) and (8.81) yields the following diffusion-type equations:
∂
u
/∂
t
)
·∇
[
ρ
a
(1
−
ρ
w
S
α
∂
+
ρ
w
Sn
0
β
w
∂
p
w
∂
t
+
ρ
w
S
α
∂
+
n
0
ρ
w
∂
S
∂
t
χ
p
w
)
)
p
a
]
∂
t
(
∂
t
[(1
−
χ
μ
w
∇
p
w
ρ
a
(1
−
S
)
α
∂
k
w
ρ
w
=∇·
∂
t
[(1
−
χ
)
p
a
]
+
ρ
a
(1
−
S
)
n
0
β
a
∂
p
a
∂
t
+
ρ
a
(1
−
S
)
α
∂
∂
t
(
χ
p
w
)
−
n
0
ρ
a
∂
S
∂
t
k
a
ρ
a
μ
a
∇
p
a
=∇·
(8.82)
in which, on account of (8.77),
α
can be defined as the vertical compressibility of the solid
frame
α
=
(2
μ
+
λ
)
−
1
(8.83)
Note that this is different from the volumetric compressibility
K
−
1
s
≡
3
e
/
τ
xx
+
τ
yy
+
τ
zz
)],
[
(
that is, the inverse of Equation (8.71).
The physical significance of the terms in Equations (8.82) can be explained as follows.
The entire left-hand side of both equations represents the local rate of change of storage of
the fluid in question at a point. In the case of the first of (8.82), which describes the flow
of the water (i.e. the wetting fluid), the first term on the left is the rate of change of storage
resulting from compression (or expansion) of the solid matrix caused by pressure changes
in the water; the second term represents the storage rate of change caused by compression
(or expansion) of the water; the third term is the rate of change of storage resulting from
bulk volume changes of the solid matrix caused by pressure changes in the air. The fourth
term shows the rate of change of water storage resulting from local changes of the degree of
saturation. Finally, the right-hand side, which is a divergence of the Darcy flux, represents
the storage rate of change as the difference between the inflow and outflow rate of water at
the point in question. The different terms of the second of (8.82), which describes the flow
of air, represent,
mutatis mutandis
, the same mechanisms as those in the first.
Before proceeding, for a better understanding of Equations (8.82) and their limitations,
the basic assumptions may be briefly repeated.
1.
The grains are incompressible.
2.
The effective stress is obtainable by means of Bishop's parameter
χ
=
χ
(
S
).
3.
The solid displacements are sufficiently small, so that the solid frame is elastic
within the range of
u
; when this is not the case, this can sometimes be remedied. For
example, as shown in Brutsaert and Corapcioglu (1976) the basic derivation leading
to Equation (8.84) can be readily extended to flow in a viscoelastic aquifer.
4.
The solid displacements are vertical only, with a constant total load. Verruijt (1969)
has described situations of saturated flow when this assumption is not valid.