Environmental Engineering Reference
In-Depth Information
Figure 7.25
Three-dimensional water flow through an unsaturated soil element.
where:
The fourth, fifth, and sixth terms in Eq. 7.41 account for
the spatial variation in the coefficient of permeability. In
the case of two-dimensional flow, the hydraulic head gradi-
ent in the third direction is assumed to be negligible (e.g.,
∂h
w
/∂z
v
wz
=
water flow rate across a unit area of the soil in the
z
-direction.
0), and Eq. 7.41 reverts to Eq. 7.36.
For the heterogeneous, isotropic case, the coefficients of
permeability in the
x
-,
y
-, and
z
-directions are equal and
Eq. 7.41 takes the following form:
k
w
∂
2
h
w
=
The above steady-state equation reduces to the following
form:
∂
v
wx
∂x
+
dx dy dz
∂
v
wy
∂y
+
∂
v
wz
∂z
=
0
(7.39)
∂z
2
∂
2
h
w
∂y
2
∂
2
h
w
∂k
w
∂x
∂h
w
∂x
∂k
w
∂y
∂h
w
∂y
+
+
+
+
∂x
2
Substituting Darcy's law into Eq. 7.39 yields the following
nonlinear partial differential equation:
∂k
w
∂z
∂h
w
∂z
+
=
0
(7.42)
k
wx
u
a
−
k
wy
u
a
−
u
w
u
w
∂
∂x
∂h
w
∂x
∂
∂y
∂h
w
∂y
+
where:
k
wz
u
a
−
u
w
∂h
w
∂z
∂
∂z
k
w
=
water coefficient of permeability in the
x
-,
y
-, and
z
-directions.
+
=
0
(7.40)
where
Table 7.6 summarizes the three-dimensional steady-state
equations for unsaturated soils. The three-dimensional
steady-state flow equations can be solved using finite
difference and finite element numerical procedures.
k
wz
(u
a
−
u
w
)
=
water coefficient of permeability as a
function of matric suction and
∂h
w
/∂z
=
hydraulic head gradient in the
z
-direc-
tion.
Table 7.6 Three-Dimensional Steady-State Equations
for Unsaturated Soils
For
the remainder of
the formulations,
the perme-
ability function terms
k
wx
u
a
−
u
w
,
k
wy
u
a
−
u
w
, and
k
wz
u
a
−
u
w
are written as
k
wx
,
k
wy
, and
k
wz
, respectively,
for simplicity. The hydraulic head distribution in a soil mass
during three-dimensional steady-state flow is described by
Eq. 7.40, which can be rewritten as follows:
Heterogeneous, anisotropic:
∂
2
h
w
∂x
2
∂
2
h
w
∂y
2
∂
2
h
w
∂z
2
∂k
wx
∂x
∂h
w
∂x
k
wx
+
k
wy
+
k
wz
+
∂k
wy
∂y
∂h
w
∂y
∂k
wz
∂z
∂h
w
∂z
∂
2
h
w
∂x
2
∂
2
h
w
∂y
2
∂
2
h
w
∂z
2
∂k
wy
∂y
∂k
wx
∂x
∂h
w
∂x
∂h
w
∂y
+
+
=
0
k
wx
+
k
wy
+
k
wz
+
+
Heterogeneous, isotropic:
k
w
∂
2
h
w
∂k
wz
∂z
∂h
w
∂z
+
=
0
(7.41)
∂z
2
∂
2
h
w
∂y
2
∂
2
h
w
∂k
w
∂x
∂h
w
∂x
∂k
w
∂y
∂h
w
∂y
+
+
+
+
∂x
2
where:
∂k
w
∂z
∂h
w
∂z
=
+
0
∂k
wz
/∂z
=
change in water coefficient of permeability in
the
z
-direction.
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