Geoscience Reference
In-Depth Information
8.7
You are given a two-dimensional flow net in a principal axes system describing the steady flow
in a homogeneous, anisotropic soil. The equipotential lines (i.e., lines of constant
h
) are a set
of equidistant straight lines making an angle of
−
20
◦
with the
x
-axis (or
+
70
◦
with the
y
-axis).
The stream lines (i.e., lines tangent to the local direction of the flow) are also a set of equidistant
straight lines, making an angle of
+
40
◦
with the
x
-axis (or
−
50
◦
with the
y
-axis). (a) Sketch
this flow net. (b) If
h
increases from bottom to top in your sketch, and if
k
xx
=
10
−
5
cm s
−
1
,
determine
k
yy
.
8.8
Derive the expression (8.41) for the soil-water diffusivity from its component functions (8.36) and
(8.15).
=
dS
e
/
dR
,
8.9
Derive the pore size probability density function,
s
e
(
R
)
implied by the soil-water
characteristic functions (8.14) and (8.15). Plot these two functions for
H
b
=
33 cm and
b
=
5
.
7,
and for
a
=
0
.
03 cm
−
1
and
b
=
5
.
7, respectively.
8.10
Extend the Richards equation (8.56) to the general case of an anisotropic material. Use
Equation (8.54) as your starting point.
8.11
Derive Equation (8.45) from (8.44).
8.12
Derive Equation (8.49) from (8.47).
8.13
(a) Apply Equation (8.49) with Ge
=
1
/
8 to calculate the intrinsic permeability at saturation
(in cm
2
) of the sandy soil, whose soil-water characteristic is shown in Figures 8.5 and 8.20.
Assume that the parameters are
θ
0
=
0
.
405
,
S
r
=
0
.
1
,
b
=
5
.
7 and
H
b
=
0
.
33 m in Equation (8.14).
(b) Calculate the saturated, hydraulic conductivity at 20
◦
C from the permeability obtained in part
(a). (Note that, experimentally, it was measured to be
k
0
=
1 cm min
−
1
, approximately.)
8.14
Prove Equation (8.70) starting from (8.67).
8.15
Derive the first of Equations (8.82) from the first of (8.74) (via the first of (8.76)).
8.16
Multiple choice. Indicate which of the following statements are correct. Hysteresis in the moisture
content-suction curve of a soil (suction is negative pressure):
(a)
is related to the geometry of the pores;
(b)
can be determined by using only a tensiometer;
(c)
is an important factor to be considered in the determination of the flow rate, when the soil
is saturated;
(d)
suggests that the flow takes place below the water table;
(e)
must be considered in the analysis of problems involving alternate wetting and drying.
8.17
Multiple choice. Indicate which of the following statements are correct. Darcy's law is not
applicable:
(a)
to describe extremely unsteady phenomena;
(b)
when the soil is non-homogeneous;
(c)
when the hydraulic head becomes large;
(d)
when the pressure is zero (i.e. atmospheric);
