Geoscience Reference
In-Depth Information
SS
WT=FS
p
w
>0
η
D
z
D
c
x=B
x
Fig. 10.11 Schematic representation of the flow domain some time after the onset of drainage in a
two-dimensional unconfined riparian aquifer, lying on a horizontal impermeable layer. The flow above
the water table (WT) is assumed negligible, so that the water table is a true free surface (FS). The
effect of capillarity is parameterized by means of the drainable porosity (or specific yield). The initial
position of the free surface is at the soil surface (SS), or
η
=
D
as shown in Figures 10.2 and 10.3.
10.2
FREE SURFACE FLOW: A FIRST APPROXIMATION
Whenever the effects of capillarity can be assumed to be relatively unimportant, the flow
in the partly saturated zone above the water table, in which
p
w
<
0, can be neglected. The
moving water table can then be treated as a true free surface, which represents the upper
boundary of the changing flow domain. As noted above, the dimensionless capillary
zone number can provide an indication of the relative importance of the capillary effects
in an unconfined aquifer. This capillary zone number is defined in Equation (10.4)
as Ca
D
, in which
D
is an average thickness of the unconfined aquifer under
consideration, and
H
c
is a characteristic capillary rise above the water table in the aquifer,
that is, a characteristic capillary suction, which reduces the degree of saturation of the
soil to a certain fraction, say 50%. Thus, whenever Ca is small, the partly saturated zone
above the water table can be eliminated from the flow domain, and the flow is assumed
to take place only below the moving water table.
=
H
c
/
10.2.1
General formulation
Differential equation and boundary conditions
Because in this approximation the flow region below the free surface is fully saturated, the
governing equation is again Laplace's Equation (10.5). For the simple two-dimensional
case of an unconfined aquifer on a horizontal bed, which is initially fully saturated, the
boundary conditions can be taken as (10.2) and (10.3) from which the partly saturated
zone is eliminated (see Figure 10.11). They can be written as follows
h
=
D
c
x
=
0
0
≤
z
≤
D
c
t
≥
0
h
=
z
x
=
0
D
c
≤
z
=
ht
≥
0
∂
h
x
=
0
x
=
B
0
≤
z
≤
Dt
≥
0
(10.13)
∂
∂
h
z
=
00
≤
x
≤
Bz
=
0
t
≥
0
∂
h
=
D
0
≤
x
≤
Bz
=
D
t
=
0
