Geoscience Reference
In-Depth Information
Fig. 10.15 Definition sketch for
two-dimensional
groundwater flow with a
water table (WT) above
a sloping impermeable
layer (IL). The WT is
assumed to be a true free
surface of the flowing
water, and flow above
the WT is neglected.
Soil surface
η
WT
IL
z
α
x
α
δζ
δ
z
δ
x
coordinate
ζ
by
∂ζ/∂
z
=
cos
α
and
∂ζ/∂
x
=
sin
α
. Integration of Equation (10.23)
yields
p
w
=
γ
w
cos
α
(
η
−
z
)
(10.24)
where
t
) is the height of the water table, again measured in the direction normal
to the impermeable bed. From Equation (10.24) it follows that, for a constant bed slope
η
=
η
(
x
,
α
,
the pressure gradient in the direction of flow
x
is given by
∂
p
w
∂
α
∂η
∂
x
=
γ
w
cos
(10.25)
x
This shows that this gradient depends only on the slope of the free surface, and is
independent of
z
; put differently,
x
is a constant along the direction normal to the
impermeable bed. For a fluid of constant density the hydraulic head is Equation (8.18)
or in the present notation
h
∂
p
w
/∂
=
ζ
+
p
w
/γ
w
. With (10.25) the hydraulic gradient becomes
∂
h
α
∂η
∂
x
=
x
+
α
cos
sin
(10.26)
∂
Hence, under the assumption of hydraulic flow, Darcy's equation yields the specific flux
k
0
cos
α
∂η
∂
q
x
=−
x
+
sin
α
(10.27)
which, as observed below Equation (10.25), is a constant in any given cross section
normal to the underlying bed at a distance
x
from the origin. A derivation of (10.27) was
first presented by Boussinesq (1877), and was later clarified by Childs (1971); however,
in both instances the approach differed somewhat from the one given here.