Environmental Engineering Reference
In-Depth Information
Let us consider steady-state seepage through an earth
dam (Fig. 8.37b). In early soil mechanics, the assumption
was generally made that there was negligible water flow
through the unsaturated soil zone due to its low coefficient
of permeability. The phreatic line was assumed to behave
as an impervious, uppermost boundary when constructing
a flownet. This uppermost boundary (i.e., line BC in
(Fig. 8.37b) is considered to be not only a phreatic line but
also an uppermost flow line. The uppermost boundary was
referred to as a free surface under these special conditions
(Freeze and Cherry, 1979). However, the position of the
free surface was unknown and its location needed to be
approximated prior to constructing the flownet.
The position of the free surface was usually determined
using an empirical procedure (Casagrande, 1937). The
assumption that the free surface is a phreatic line requires
that the pore-water pressures be zero along this line. Equipo-
tential lines must intersect the free surface (which is also
an uppermost flow line) at right angles. In other words, it
was assumed that there was no flow across the free surface.
The flownet could then be drawn.
The flownet technique was mainly suitable for analyz-
ing steady-state seepage through isotropic, homogeneous,
saturated soils. The flownet technique becomes complex
and difficult to use when analyzing anisotropic, heteroge-
neous soil systems. There is an inherent problem associated
with applying the flownet technique to saturated-unsaturated
flow. Freeze (1971) stated that “the boundary conditions that
are satisfied on the free surface specify that the pressure head
must be atmospheric and the surface must be a streamline.
Whereas the first of these conditions is true, the second
is not.” The incorrect assumption regarding the uppermost
boundary condition can be avoided by recognizing that there
can be water flow between the saturated and unsaturated
zones (Freeze, 1971; Papagiannakis and Fredlund, 1984).
Steady-state flow in the saturated and unsaturated zones
can be solved simultaneously using the general partial dif-
ferential equation governing water flow through saturated-
unsaturated soils. Both zones are treated as a single domain.
The water coefficient of permeability in the saturated zone
is equal to k s . The water coefficient of permeability k w
varies with respect to the matric suction in the unsaturated
zone. The flownet technique is not applicable to saturated-
unsaturated flow modeling since the governing flow equation
is not of the Laplacian form. The general flow equation can
be solved using a numerical technique such as the finite dif-
ference method or the finite element method. Figure 8.38
shows several typical solutions by Freeze (1971) involving
saturated-unsaturated flow modeling.
Figure 8.38 Typical solutions for saturated-unsaturated flow
modeling of various dam sections (after Freeze, 1971).
two-dimensional problems. Figure 8.39 shows the cross
section of a dam that has been subdivided into triangular ele-
ments. The lines separating the elements intersect at nodal
points. The hydraulic head at each nodal point is obtained
by solving the governing flow equation for the applied
boundary conditions.
The finite element formulation for steady-state seepage in
two dimensions has been derived using the Galerkin prin-
ciple of weighted residuals (Papagiannakis and Fredlund,
1984):
T
∂x {
k wx
L
}
0
∂y {
0
k wy
A
L
}
8.3.5 Steady-State Seepage Analysis Using Finite
Element Method
The application of the finite element method requires the
discretization of the soil mass into elements. Triangular and
quadrilateral shapes of elements are commonly used for
∂x {
dA h wn
L
}
T
×
S {
L
}
v w dS p =
¯
0
(8.44)
∂y {
L
}
 
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