Environmental Engineering Reference
In-Depth Information
and is called the minor coefficient of permeability
k
w
2
.The
ratio of the major to the minor coefficients of permeability
is a constant greater than unity at any point within the soil
mass. The magnitudes of the major and minor coefficients
of permeability,
k
w
1
and
k
w
2
, can vary with matric suction
from one location to another (i.e., heterogeneity), but their
ratio is assumed to remain constant at every point.
An unsteady-state seepage formulation and subsequent
solution are generally performed with respect to the
x
and
y
Cartesian coordinate directions. Therefore, it is necessary
to write the coefficients of permeability for the
x
- and
y
-
directions in terms of the major and minor coefficients of
permeability. The relationship between the two permeabil-
ity systems can be derived by first writing the water flow
rates in the major and minor permeability directions (i.e.,
directions
s
1
and
s
2
, respectively):
where:
∂h
w
/∂x
=
hydraulic head gradient in the
x
-direction and
∂h
w
/∂y
=
hydraulic head gradient in the
y
-direction.
From trigonometric relations, the following relationships
can be obtained (see Fig. 7.26):
dx
ds
1
=
cos
α
(7.47)
dy
ds
1
=
sin
α
(7.48)
dx
ds
2
=−
sin
α
(7.49)
dy
ds
2
=
cos
α
(7.50)
∂h
w
∂s
1
v
w
1
=−
k
w
1
(7.43)
Substituting Eqs. 7.45-7.50 into Eqs. 7.43-7.44 gives
∂h
w
∂s
2
k
w
1
cos
α
∂h
w
∂x
v
w
2
=−
k
w
2
(7.44)
sin
α
∂h
w
∂y
v
w
1
=−
+
(7.51)
k
w
2
where:
sin
α
∂h
w
∂x
cos
α
∂h
w
∂y
v
w
2
=−
−
+
(7.52)
v
w
1
=
water flow rate across a unit area of the soil
element in the
s
1
-direction,
The water flow rates in the
x
- and
y
-directions can be
written by projecting the flow rates in the major and minor
directions to the
x
- and
y
-directions:
v
w
2
=
water flow rate across a unit area of the soil
element in the
s
2
-direction,
k
w
1
=
major coefficient of permeability with respect
to water as a function of matric suction which
v
wx
=
v
w
1
cos
α
−
v
w
2
sin
α
(7.53)
varies in the
s
1
-direction [i.e.,
k
w
1
u
a
−
u
w
],
v
wy
=
v
w
1
sin
α
+
v
w
2
cos
α
(7.54)
k
w
2
=
minor coefficient of permeability with respect
to water as a function of matric suction which
varies in the
s
2
-direction [i.e.,
k
w
2
u
a
−
u
w
],
where:
h
w
=
hydraulic head (i.e., gravitational plus pore-
water pressure head, or
y
v
wx
=
water flow rate across a unit area of the soil element
in the
x
-direction,
+
u
w
/ρ
w
g
),
y
=
elevation,
v
wy
=
water flow rate across a unit area of the soil element
in the
y
-direction, and
s
1
=
direction of major coefficient of permeability,
k
w
1
,
α
=
angle between the direction of the major coefficient
of permeability and the
x
-direction.
s
2
=
direction of minor coefficient of permeability,
k
w
2
,
∂h
w
/∂s
1
=
hydraulic head gradient in the
s
1
-direction,
and
Substituting Eqs. 7.51 and 7.52 for
v
w
1
and
v
w
2
, respec-
tively, into Eqs. 7.53 and 7.54 results in the following
relations:
∂h
w
/∂s
2
=
hydraulic head gradient in the
s
2
-direction.
The chain rule can be used to express the hydraulic head
gradients in the
s
1
- and
s
2
-directions (i.e.,
∂h
w
/∂s
1
and
∂h
w
/
∂s
2
, respectively) in terms of the gradients in the
x
- and
y
-directions (i.e.,
∂h
w
/∂x
and
∂h
w
/∂y
, respectively):
k
w
1
cos
2
α
∂h
w
∂x
k
w
1
sin
α
cos
α
∂h
w
∂y
v
wx
=−
−
k
w
2
sin
2
α
∂h
w
k
w
2
sin
α
cos
α
∂h
w
∂y
−
∂x
+
(7.55)
∂h
w
∂s
1
=
∂h
w
∂x
∂x
∂s
1
+
∂h
w
∂y
∂y
∂s
1
k
w
1
sin
α
cos
α
∂h
w
∂x
k
w
1
sin
2
α
∂h
w
∂y
(7.45)
v
wx
=−
−
∂h
w
∂s
2
=
∂h
w
∂x
∂x
∂s
2
+
∂h
w
∂y
∂y
∂s
2
k
w
2
sin
α
cos
α
∂h
w
∂x
k
w
2
cos
2
α
∂h
w
∂y
(7.46)
+
−
(7.56)
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