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which is covered in Section 10.3. Linearization of the hydraulic groundwater approach
constitutes a third approximate formulation, and this is covered in Section 10.4. Finally,
the additional assumption that the hydraulic head gradient is equal to the slope of the
land surface produces a kinematic wave formulation, which is the fourth approximation,
treated in Section 10.5. But before looking more closely into these common simplified
approaches in the remainder of this chapter, it is useful first to consider briefly some
implications of solutions of Equation (10.1) itself.
10.1.3
A few features of combined saturated-unsaturated flow
Unsteady flow formulation
Some results of a numerical solution of Equation (10.1) were presented by Verma and
Brutsaert (1970; 1971b) for the two-dimensional case of outflow from a horizontal
unconfined aquifer with a rectangular cross section, after the cessation of recharge; this
situation is illustrated in Figure 10.2. The soil water characteristic was assumed to be
given by Equation (8.15) and the hydraulic conductivity by Equation (8.36). With the
effective saturation S e defined in Equation (8.6), the boundary conditions for this problem
can be specified as follows
h
=
D c
S e =
1
.
0
x
=
0
0
z
D c
h
=
z
e =
1
.
0
x
=
0
D c
z
=
h
h
S e
x =
0
x =
0
x
=
0
h
z
D
h
S e
(10.2)
x =
0
x =
0
x
=
B
0
z
D
h
S e
z =
0
z =
00
x
B
z
=
0
h
S e
z =
0
z =
00
x
Bz
=
D
The first boundary condition is a result of the hydrostatic pressure distribution in the
stream. At the seepage surface the pressure is zero (i.e. atmospheric) and the hydraulic
head is equal to the height z , as indicated in the second condition. Above the seepage
surface the water pressure is negative, and because no outflow is physically possible
unless the pressure is at least atmospheric, this surface acts as an impermeable boundary
as indicated in the third of (10.2). The boundary conditions given by the fourth and fifth
of (10.2) express the no-flow or impermeable boundaries of the aquifer at the divide and
at the underlying bed rock. As it is assumed that there is no evaporation or recharge, the
ground surface acts like an impermeable boundary after the drainage starts, as indicated
in the last condition; however, this condition at the ground surface can be readily replaced
by the evaporative flux or recharge rate, if it is known.
The initial conditions, for t
0, can be assumed to be those of a fully saturated
aquifer, in which the water table coincides with the ground surface. This situation is
=
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