Geoscience Reference
In-Depth Information
The normal strain is defined as the change in length of an element in a certain direction
as a result of deformation, divided by the original length, in the limit for an infinitesi-
mally small element. In the
x
-direction, the original length is
δ
x
and the deformed length
(
x
F
−
x
H
), which yields immediately
e
xx
=
∂
u
x
/∂
x
; the same reasoning,
mutatis mutandis
,
produces
e
zz
and
e
yy
. Note that the reasoning is similar to the derivation of Equations (1.5)
and (1.6). The sum of the normal strains, which equals the fractional change in volume of
the deformed cube of solid skeleton, is the volume strain, also called the dilatation, i.e.,
e
=∇·
u
=
e
xx
+
e
yy
+
e
zz
(8.58)
The shear strain is by definition one half the change in angle between two originally per-
pendicular elements, as deformation takes place, again in the limit for an infinitesimally
small element. In the case of A
B
C
D
the shear strain is one half the sum of the angle of
HF with the
x
-axis and the angle of EG with the
z
-axis; the angle of HF with the
x
-axis is
(
z
F
−
z
H
)
/δ
x
and the angle of EG with the
z
-axis is (
x
G
−
x
E
)
/δ
z
, so that the (
xz
)-component
of the shear strain is
e
xz
=
e
zx
=
(
∂
u
x
/∂
z
+
∂
u
z
/∂
x
)
/
2; the two other shear strain compo-
nents
e
xy
and
e
yz
are obtained in a similar way.
The relevant strains for the fluids are the changes in volume of fluid per unit bulk volume
of porous material, that is, for the water,
e
w
=∇ ·
w
=
∂w
x
∂
x
+
∂w
y
∂
y
+
∂w
z
(8.59)
∂
z
and similarly, the dilatation of the air,
e
a
=∇ ·
v
=
∂v
x
∂
x
+
∂v
y
∂
y
+
∂v
z
(8.60)
∂
z
Because the fluid displacements represent fluid volume per unit bulk area of porous material,
the corresponding changes in volume of fluid per unit volume of fluid are
e
w
/
(
n
0
S
) and
e
a
/
[
n
0
(1
−
S
)], respectively.
The displacements can readily be shown to satisfy the following equations of continuity,
in accordance with Equation (1.8); namely, for the water:
∂
∂
t
(
ρ
w
n
0
S
)
=−∇·
ρ
w
∂w
∂
t
(8.61)
for the air,
∂
∂
t
[
ρ
a
n
0
(1
−
S
)]
=−∇·
ρ
a
∂
v
∂
t
(8.62)
and for the solid,
(1
−
n
0
)
∂
u
∂
t
∂
n
0
∂
t
=∇ ·
(8.63)
in which
ρ
w
and
ρ
a
are the density of the water and of the air, respectively,
n
0
is the porosity,
and
S
=
θ/
n
0
is the degree of saturation of the material with water, that is, the volume of
water per unit volume of pore space and
θ
the volumetric water content. Observe that
Equation (8.63) is based on the assumption that the density of the solid phase (namely, the
grains but not the solid frame) is constant.
