Environmental Engineering Reference
In-Depth Information
In the case of a
heterogeneous
,
saturated
soil, the coeffi-
cients of permeability can be replaced by
k
s
in Eq. 8.41:
a soil mass. The solution can be obtained using closed-form
analytical methods, analog methods, or numerical methods.
Historically, a graphical method referred to as drawing
a
flownet
has been used to solve the Laplacian equation
(Casagrande, 1937).
The flownet solution results in two families of curves
referred to as flow lines and equipotential lines. The flownet
solution has been used extensively to analyze problems
involving seepage through saturated soils and is explained
in most soil mechanics textbooks. Boundary conditions for
the soil domain must be known prior to the construction
of the flownet. Either a head or zero flux is prescribed
along the boundary. A boundary condition exception is
the case of a free surface. A network of flow lines and
equipotential lines is sketched by trial and error in order
to satisfy the boundary conditions and the requirement of
right-angled, equal-dimensional elements.
A hydraulic head boundary condition or an imperme-
able boundary condition can readily be imposed for most
saturated soil problems. For example, steady-state seepage
beneath a sheet pile wall has the boundary conditions shown
in Fig. 8.37a. However, the boundary conditions can be more
difficult to assign when dealing with unsaturated soils.
4
k
s(i)
+
2
y
A
k
s(i
+
1
)
−
k
s(i
−
1
)
h
w
(i)
=
h
w
(i
−
1
)
−
q
wy
8
k
s(i)
(8.42)
Equation 8.42 becomes linear when the soil is homoge-
neous:
y
Ak
s
h
w
(i)
=
h
w
(i
−
1
)
−
q
wy
(8.43)
Equation 8.43 specializes to a linear distribution of the
hydraulic head for a homogeneous, saturated soil column
subjected to one-dimensional steady-state flow.
8.3.4 Solution of Two-Dimensional Water Flow
Problems
Two-dimensional steady-state flow through a homogeneous,
isotropic saturated soil must have the
x-
and
y-
direction
components added together. The result is a partial differ-
ential equation referred to as the Laplacian equation. The
solution of this equation describes the head at all points in
Sheet pile wall
H
1
H
2
B
A
Datum
F
E
Saturated
soil
CD
H
G
Impervious
(a)
Assumed impervious
Reservoir level
B
Free surface and
uppermost flowline
H
1
Flow lines
Saturated soil
Datum
A
C
D
Impervious
Boundary conditions:
AB
:
h
w
=
H
1
BC
: free surface, its location is unkown
CD
:
h
w
= 0
DA
:
q
w
= 0
Equipotential lines
Horizontal drain
(b)
Figure 8.37
Flownet constructions to solve the Laplacian partial differential equation:
(a) steady-state seepage throughout homogeneous, isotropic saturated soil, (b) steady-state seepage
throughout homogeneous, isotropic earth dam.
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