Civil Engineering Reference
In-Depth Information
(Muskat, 1937) is,
2
2
k
x
∂
φ
∂x
+
k
y
∂
φ
=
0
(2.122)
2
2
∂y
where
φ
is the fluid “potential” or total head and
k
x
and
k
y
are permeabilities or con-
ductivities in the
x
and
y
directions. The finite element discretisation process reduces the
differential equation to a set of equilibrium type matrix equations of the form,
[
k
c
]
{
φ
} = {
q
}
(2.123)
where
[
k
c
] is the symmetrical “conductivity matrix”,
{
φ
}
is a vector of nodal potential
(total head) values, and
is a vector of nodal inflows/outflows.
With the usual finite element discretisation,
{
q
}
φ
=
[
N
]
{
φ
}
(2.124)
reference to Table 2.1 shows that typical terms in the matrix [
k
c
] are of the form,
d
x
d
y
k
x
∂N
i
∂x
∂N
j
∂x
+
k
y
∂N
i
∂y
∂N
j
∂y
(2.125)
A convenient way of expressing the matrix
[
k
c
] in (2.123) is,
[
T
]
T
[
K
][
T
] d
x
d
y
[
k
c
]
=
(2.126)
where the property matrix [
K
] is analogous to the stress-strain matrix [
D
] in solid mechan-
ics, where,
k
x
0
[
K
]
=
(2.127)
0
k
y
(assuming that the principal axes of the permeability tensor coincide with
x
and
y
). The
[
T
] matrix is similar to the [
B
] matrix of solid mechanics and is given by (for a 4-node
element),
∂N
3
∂x
∂N
1
∂x
∂N
2
∂x
∂N
4
∂x
[
T
]
=
(2.128)
∂N
1
∂y
∂N
2
∂y
∂N
3
∂y
∂N
4
∂y
The similarity between (2.126) for a fluid and (2.69) for a solid enables the correspond-
ing programs to look similar in spite of the governing differential equations being quite
different. This unity of treatment is utilised in describing the programming techniques in
Chapter 3.
Finally, it is worth noting that (2.126) can also be arrived at from energy considerations.
The equivalent energy statement is that the integral
1
2
k
x
2
d
x
d
y
∂φ
∂x
2
∂φ
∂y
1
2
k
y
+
(2.129)
shall be a minimum for all possible
φ(x,y)
.
