Environmental Engineering Reference
In-Depth Information
Equations 8.34 and 8.35 are called the central-difference
approximations for the first and second derivatives of the
function
h(y)
at point
i
. These approximations can be used
to solve the differential seepage equation. Similar approxi-
mations can be derived for a function
h(x)
in the
x
-direction
when there is flow in the
x-
direction. The use of an iterative
finite difference technique in solving seepage problems is
illustrated in the following sections.
8.3.2 Head Boundary Conditions Applied
Steady-state evaporation under a fixed head boundary
condition from a column of unsaturated soil is illus-
trated in Fig. 8.33. A tensiometer is installed near the
ground surface to measure the negative pore-water pressure.
One-dimensional, steady-state flow is assumed when the ten-
siometer reading remains constant with respect to time. The
pore-water pressure at the base of the column (i.e., the water
table) is equal to zero.
The hydraulic head distribution along the
Figure 8.32
Hydraulic head function
h(y)
written in finite differ-
ence form.
can be used to illustrate the solution to the steady-state
flow equation for an unsaturated soil.
The seepage differential equation can be written in a finite
difference form. Consider the situation where a function
h(y)
varies in space, as shown in Fig. 8.32. Values of the
function at points along the curve describing hydraulic head
can be computed using a Taylor series to write forward-
difference and backward-difference equations. The forward
difference-equation can be written as
length
of
the column is given by Eq. 7.30 [Chapter 7;
i.e.,
k
wy
d
2
h
w
dy
2
+
dk
wy
dy
dh
w
dy
=
0]. This equation
can be solved using the finite difference approximations
shown in Eqs. 8.34 and 8.35. The column length is first
subdivided into
n
equally spaced nodal points at a distance
y
apart (Fig. 8.33). A central-difference approximation
is then applied to the hydraulic head and coefficient-
of-permeability derivatives. The finite difference form can
be written as follows for point
i
:
y
dh
dy
d
2
h
dy
2
y
2
2!
h
i
+
1
=
h
i
+
i
+
h
w
(i
+
1
)
+
i
h
w
(i
−
1
)
−
2
h
w
(i)
d
3
h
dy
3
y
2
y
3
3!
k
wy
(i)
+
i
+···
(8.32)
k
wy
(i
+
1
)
−
h
w
(i
+
1
)
−
k
wy
(i
−
1
)
2
y
h
w
(i
−
1
)
2
y
+
=
0 (8.36)
The backward-difference equation can be written as
y
dh
dy
d
2
h
dy
2
y
2
2!
where:
h
i
−
1
=
h
i
−
i
+
i
d
3
h
dy
3
k
wy
(i)
,
k
wy
(i
−
1
)
,
k
wy
(i
+
1
)
=
water coefficients of perme-
ability in the
y
-direction at
points
i
,
i
y
3
3!
−
i
+···
(8.33)
−
1, and
i
+
1, re-
where:
i
spectively, and
−
1,
i
,
i
+
1
=
three consecutive points spaced at incre-
h
w
(i)
,
h
w
(i
−
1
)
,
h
w
(i
+
1
)
=
hydraulic heads at points
i
,
i
−
ments
y
.
Subtracting the forward- and backward-difference
equations and neglecting the higher order derivatives result
in the first derivative of the function at point
i
:
dh
dy
1, and
i
+
1, respectively.
Equation 8.36 can be rearranged after assuming equal
y
increments:
−
8
k
wy
(i)
·
h
w
(i)
+
4
k
wy
(i)
+
k
wy
(i
−
1
)
·
h
i
+
1
−
h
i
−
1
2
y
k
wy
(i
+
1
)
−
h
w
(i
+
1
)
i
=
(8.34)
+
4
k
wy
(i)
+
k
wy
(i
+
1
)
·
k
wy
(i
−
1
)
−
h
w
(i
−
1
)
=
0
(8.37)
Summing the forward- and backward-difference equations
and again neglecting the higher order derivatives give the
second derivative of the function at point
i
:
d
2
h
The hydraulic heads at the external points (i.e., points
1 and
n
) become the boundary conditions. The hydraulic
head at point 1 is zero. The elevation of point
n
relative
to the datum,
h
gn
, gives the gravitational head at point
n
.
The tensiometer reading near the ground surface indicates
dy
2
h
i
+
1
+
h
i
−
1
−
2
h
i
i
=
(8.35)
y
2
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