Environmental Engineering Reference
In-Depth Information
L T λ L dA h wn
∂t
B T k w B dA h wn +
A
A
[ L ] T
¯
v w dS
=
0
(8.51)
S
where:
[ B ]
=
matrix of the derivatives of the area coordinates,
as shown for the steady-state formulation,
[ L ]
=
matrix of
the element area coordinates
(i.e.,
L 1 L 2 L 3 ), and
ρ w gm 2 .
λ
=
Boundary conditions must also be defined for an
unsteady-state finite element analysis. Either the hydraulic
head or the flow rate must be specified at the boundary
nodal points. Specified hydraulic heads at the boundary
nodes are called Dirichlet boundary conditions. A specified
flow rate across the boundary is referred to as a Neuman
boundary condition. The third term in Eq. 8.51 (i.e., the
terms just before the equal sign) accounts for the specified
flow rates across the boundary. The specified flow rates at
the boundary must be projected to a direction normal to the
boundary. As an example, a specified flow rate v w in the
vertical direction must be converted to a normal flow rate
Figure 8.50 Effect of slope inclination on the pore-water pressure
distribution along a vertical plane.
The pore-water pressure head equation indicates that
there is a decrease in the hydraulic head as the datum is
approached. In other words, there is a vertical downward
component of water flow. The above analysis also applies to
the pore-water pressure conditions below the phreatic line.
Using the same horizontal line through point A , positive
pore-water pressure heads along a vertical plane can be
computed in accordance with Eq. 8.49. The hydraulic head
(Eq. 8.50) is zero at the phreatic line and decreases linearly
with depth along a vertical plane.
v w .
The normal flow rate must in turn be converted to a nodal
flow Q w (Segerlind, 1984). A positive nodal flow signifies
that there is infiltration at the node or that the node acts
as a source. A negative nodal flow indicates evaporation or
evapotranspiration at the node and shows that the node acts
as a sink. When the flow rate across a boundary is zero (e.g.,
impervious boundary), the third term in Eq. 8.51 disappears.
The numerical integration of Eq. 8.51 results in the
following expression for the saturated-unsaturated seepage
equation:
¯
8.3.8 Examples of Two-Dimensional, Unsteady-State
Water Flow Problems
Example problems are presented to illustrate unsteady-state
seepage solutions when using the finite element method. The
finite element formulation for two-dimensional, saturated-
unsaturated seepage analyses is first presented followed
by the solution of several example problems. Unsteady-
state problems require that the permeability function and
the water storage function (i.e., k w and m 2 ) be known
for each soil strata that might become desaturated at any
elapsed time. Both the permeability function and the water
storage function take the form of a nonlinear mathematical
function for unsaturated soils.
[ D ] h wn +
[ E ] h wn =
[ F ]
(8.52)
where:
stiffness matrix, that is, B T k w B A,
[ D ]
=
211
121
112
λA
12
,
[ E ]
=
capacitance matrix, that is,
h wn =
matrix of the time derivatives of the hydraulic
heads at the nodal points (i.e., h wn /∂t ), and
[ F ]
=
flux vector reflecting the boundary conditions
(i.e., S [ L ] T
¯
v w dS ).
8.3.9 Unsteady-State Seepage Analysis Using Finite
Element Method
The finite element formulation for unsteady-state seepage in
two dimensions can be derived using the Galerkin princi-
ple of weighted residuals (Lam et al., 1987). The Galerkin
solution to the governing seepage equation is given by the
following integrals over the area and the boundary surface
of a triangular element:
The time derivative in Eq. 8.52 can be approximated using
a finite difference form. The relationship between the nodal
heads of an element at two successive time steps t can be
expressed using the central-difference approximation
[ D ]
h wn t + t =
2 [ E ]
t
[ D ] h wn t +
2 [ E ]
t
+
2 [ F ]
(8.53)
 
Search WWH ::




Custom Search