Geoscience Reference
In-Depth Information
x
(cm)
10
30
40
50
60
70
72
75
76
θ
0.45
0.45
0.45
0.44
0.42
0.36
0.33
0.20
0.10
Calculate
D
w
=
D
w
(
θ
) (in cm
2
min
−
1
) for
θ
=
0.2, 0.3, 0.4, 0.45 by solving Equation (9.25)
graphically or numerically.
9.16
Consider the horizontal infiltration experiment of Problem 9.15, which was allowed to run for
t
=
θ
=
θ
(
x
)
, that would be observed, if the
740 min. Tabulate the water content distribution
experiment were allowed to run for only
t
=
370 min.
9.17
Derive the expression for the sorptivity (9.28) from the exact solution (9.27). From this, derive an
expression for the horizontal infiltration rate
f
.
9.18
Derive (9.39) from (9.38). Hint: use integration by parts, and follow up with Leibniz's rule (see
Appendix).
9.19
A fairly accurate solution of the sorption problem is Equation (9.38) with
b
=
0
.
(It is not as accurate
as (9.43), but it is easier to handle analytically.) Use this solution to calculate the sorptivity
A
0
by
means of (9.39) and also the position of the wetting front
φ
f
; give a simple relationship between
F
and
x
f
,
as a function of
β,
if the diffusivity is given by (8.39). Compare with the more accurate
result given in (9.48), and also presented in Figure 9.12, for the values
β
=
3 and 8.
9.20
Derive Equation (9.104) from (9.103).
Consider a homogeneous sandy soil profile, whose conductivity is given by (8.37) in cm d
−
1
9.21
,
with
a
=
170
×
10
6
,
b
=
2
.
5
×
10
6
and
c
=
4
.
Assume a potential evaporation of 0
.
4cmd
−
1
from
the bare soil surface; what is the smallest depth of the water table for which the soil (instead of
the atmosphere) totally controls the evaporation?
9.22
Consider the Diablo loam, whose hydraulic conductivity is depicted in Figure 8.29. Use the values
of the parameters
a
,
b
and
c
in Equation (8.37) from the figure, and calculate the maximal rate
of evaporation (by steady capillary rise) from a bare soil surface for the following three cases; the
water table is (a) at 0.5 m, (b) at 1.0 m and (c) at 1.5 m below the surface.
9.23
Multiple choice. Indicate which of the following statements are correct. The motion of a wetting
fluid in two-phase immiscible flow problems is often described by Richards's equation. This
formulation requires the following assumptions:
(a)
Darcy's law is valid;
(b)
conservation of thermodynamic energy;
(c)
the non-wetting fluid is inviscid, so that it moves freely without pressure gradient;
(d)
The effects of capillarity are negligible;
(e)
The porous matrix is incompressible.
(9.24)
Multiple choice. Indicate which of the following statements are correct. During vertical infiltration
of ponded water (the water is maintained as a very thin layer of constant thickness) into a deep
homogeneous dry soil:
