Environmental Engineering Reference
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or the backward-difference approximation
solving highly nonlinear PDEs. The design of the finite
element mesh becomes dynamic, changing as necessary to
meet the requirements for convergence to a reliable solution.
The unsteady-state seepage equation is assumed to be
solved for one time step once the converged nodal hydraulic
heads of the system have been obtained. Having reached
convergence at a particular time step, other secondary quan-
tities, such as pore-water pressures, hydraulic head gradi-
ents, and water flow rates, can then be calculated using the
converged nodal hydraulic heads. The equation for nodal
pore-water pressures is
[
D
]
h
wn
t
+
t
=
t
h
wn
t
+
[
E
]
t
[
E
]
+
[
F
]
(8.54)
The above time derivative approximations are considered
to be unconditionally stable. The central-difference approx-
imation generally gives a more accurate solution than that
obtained from the backward-difference approximation.
However, the backward-difference approximation is found
to be more effective in reducing numerical oscillations
commonly encountered in highly nonlinear systems of flow
equations (Neuman and Witherspoon, 1971; Neuman, 1973).
The finite element seepage equation can be written for
each element and assembled to form a set of global flow
equations. The finite element formulation is solved while
satisfying nodal compatibility (Desai and Abel, 1972). Nodal
compatibility requires that a particular node shared by the
surrounding elements has the same hydraulic head in all
shared elements (Zienkiewicz, 1971; Desai, 1975b).
The global flow equations for the whole system are solved
u
wn
=
h
wn
−
y
n
ρ
w
g
(8.55)
where:
u
wn
=
matrix of pore-water pressures at
the nodal
⎧
⎨
⎫
⎬
⎛
⎞
u
w1
u
w2
u
w3
⎝
i.e.
,
⎠
,
points
⎩
⎭
y
n
=
matrix of elevation heads at the nodal points
⎛
⎧
⎨
⎫
⎬
⎞
for the hydraulic heads at the nodal points,
h
wn
. However,
Eq. 8.52 is nonlinear because the coefficients of permeability
are a function of matric suction, which is related to the
hydraulic head at the nodal points.
The hydraulic heads are unknown variables in Eq. 8.52.
The equation must be solved using an iterative procedure
that involves a series of successive approximations. For the
first approximation, the coefficients of permeability are esti-
mated in order to calculate the first set of hydraulic heads
at the nodal points. The computed hydraulic heads are used
to calculate the average matric suction within an element.
In the subsequent approximations, the coefficient of perme-
ability is adjusted to a value corresponding to the computed
average matric suction in the element. The adjusted per-
meability value is then used to calculate a new set of nodal
point hydraulic heads. The above procedure is repeated until
both the hydraulic head and the permeability differences for
each element are smaller than a specified tolerance between
two successive iterations.
The above iterative procedure allows the global flow
equations to be solved using a Gaussian elimination tech-
nique. The convergence on hydraulic head and coefficient of
permeability is dependent on the degree of nonlinearity
of the permeability function and the spatial discretization
of the continuum. A steep permeability function generally
requires an increased number of iterations to achieve
convergence. A finer discretization in both element size and
time step assists in obtaining more rapid convergence with
a smaller tolerance. The solution often converges to within
a tolerance of less than 1% in a few iterations provided the
permeability and water storage functions are not too highly
nonlinear. The use of PDE solvers with optimized mesh
design and automatic mesh refinement is of great value in
y
1
y
2
y
3
⎝
i.e.
,
⎠
.
⎩
⎭
The hydraulic head gradients in the
x
- and
y
-directions
can be computed for an element by taking the derivative
of the element hydraulic heads with respect to
x
and
y
,
respectively:
i
x
i
y
[
B
]
h
wn
=
(8.56)
where:
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
(8.57)
where:
v
wx
,
v
wy
=
water flow rates within an element in the
x
-
and
y
-directions, respectively.
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 respective elemen-
tal areas.
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