Environmental Engineering Reference
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where:
solved directly starting from a known boundary condition
such as point 1 (Fig. 8.35), which has a zero hydraulic head.
The base of the soil column is a suitable point to commence
solving for the remaining hydraulic heads. Hydraulic heads
can subsequently be solved point by point up to the ground
surface. Equation 8.41 is nonlinear since the coefficient
of permeability
k
wy
varies with the pore-water pressure
component of hydraulic head
h
w
. The equation must be
solved iteratively with the coefficient of permeability set
to constant values that are consistent with the computed
pore-water pressures.
The coefficient of permeability at each node,
k
wy
,
is
assumed to be equal to the saturated coefficient of perme-
ability,
k
s
,
for the first iteration. The computed hydraulic
heads, and subsequently the negative pore-water pressures,
are used to revise the coefficients of permeability for the
second iteration. The iterative procedure can be repeated
until there is convergence with respect to the hydraulic
heads and the coefficients of permeability. When computing
the hydraulic head at the ground surface, the
k
wy
(n
+
1
)
value
can be assumed to be equal to the
k
wy
(n)
value.
Typical distributions of pore-water pressure and hydraulic
head during steady-state infiltration are illustrated in
Fig. 8.36. The nonlinear distribution of the pore-water
pressure and hydraulic head is produced by the nonlinearity
of the differential seepage equation. The equipotential lines
are not uniformly distributed along the soil column. The
above analysis is also applicable to steady-state, upward
flow (e.g., evaporation from ground surface), where the flux
q
wy
is known.
q
wy
=
water flux through the soil column during the
steady-state flow, where flux is assumed to be
positive in an upward direction and negative in a
downward direction, and
A
=
cross-sectional area of the soil column.
Equation 8.38 can be rearranged as follows:
2
y
Ak
wy
(i)
h
w
(i
+
1
)
=
h
w
(i
−
1
)
−
q
wy
(8.39)
Substituting
h
w
(i
+
1
)
into the flow equation for point
i
yields the following form:
−
8
k
wy
(
i
)
h
w
(
i
)
+
4
k
wy
(
i
)
+
k
wy
(
i
+
1
)
h
w
(i
−
1
)
−
k
wy
(i
−
1
)
2
y
Ak
wy
(
i
)
−
q
wy
+
4
k
wy
(i)
+
k
wy
(i
+
1
)
·
k
wy
(i
−
1
)
−
h
w
(i
−
1
)
=
0
(8.40)
The hydraulic head can now be solved for point
i
:
4
k
wy
(i)
+
2
y
A
k
wy
(i
+
1
)
−
k
wy
(i
−
1
)
h
w
(i)
=
h
w
(i
−
1
)
−
q
wy
8
k
wy
(i)
(8.41)
The finite difference hydraulic head equation (i.e.,
Eq. 8.41) is in an explicit form. The hydraulic heads can be
q
wy
h
gn
h
pn
h
wn
0.9
h
wn
0.8
h
wn
0.7
h
wn
0.6
h
wn
0.5
h
wn
0.4
h
wn
0.3
h
wn
Pore-water
Pressure head,
h
p
Hydraulic head,
h
w
h
gn
Gravitational
head,
h
g
0.2
h
wn
0.1
h
wn
y
Water
table
Datum
0
(-)
0
Head,
h
(+)
q
wy
Figure 8.36
Steady-state infiltration through an unsaturated soil with a designated head at the
ground surface.
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