Environmental Engineering Reference
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where:
a dam using a saturated-unsaturated finite element seepage
analysis. The cross section and discretization of the problem
are illustrated in Fig. 8.39. The discretization of the soil con-
tinuum is designed in an optimum manner. In other words,
there are areas where it is advantageous to have smaller
elements and there are other areas where larger elements
are adequate. The nonlinear soil properties also influence
the design of the finite element mesh. An automatic finite
element mesh design algorithm has been used for all finite
element example problems shown in this topic (M.D. Fred-
lund, 2005). Consequently, each example problem has its
own finite element mesh.
Water 10 m high is applied to the upstream of the dam.
The permeability function used in the analysis is shown as
function
A
in Fig. 8.42. The saturated coefficient of per-
meability
k
s
i
x
,i
y
=
hydraulic head gradient within an element in the
x
- and
y
-directions, respectively.
The element flow rates
v
w
can be calculated from the
hydraulic head gradients and the coefficients of permeability
in accordance with Darcy's law:
v
wx
v
wy
=
k
w
[
B
]
h
wn
(8.48)
where:
v
wx
,
v
wy
=
water flow rates within an element in the
x
-
and
y
-directions, respectively.
10
−
7
m/s. The pore-air pressure is
assumed to be atmospheric. Therefore, the matric suction
values in Fig. 8.42 are numerically equal to the pore-water
pressures. The base of the dam is chosen as the datum.
The first example is an isotropic earth dam with a hori-
zontal drainage layer near the downstream toe, as shown in
Fig. 8.43. The 10-m-high water on the upstream of the dam
gives a 10-m hydraulic head at each node along the upstream
face. A zero hydraulic head is specified at nodes along the
horizontal drain. Zero nodal flow is specified at nodes along
the remaining boundaries. The results of the finite element
analysis are presented in Figs. 8.43a and 8.43b.
is 1
.
0
×
The hydraulic head gradient and the flow rate at nodal
points are computed by averaging the corresponding quan-
tities from all elements surrounding the node. The weighted
average is computed in proportion to the element areas.
8.3.6 Examples of Two-Dimensional, Steady-State
Water Flow Problems
The following examples are presented to demonstrate the
application of the finite element method to steady-state seep-
age through saturated-unsaturated soils. Papagiannakis and
Fredlund (1984) solved several examples of seepage through
10
−
6
10
−
7
Funtion A
10
−
8
10
−
9
10
−
10
10
−
11
Funtion B
10
−
12
10
−
13
0
1
10
100
1000
Soil suction, kPa
Figure 8.42
Specified permeability functions for analyzing steady-state seepage through earthfill
dam.
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