Environmental Engineering Reference
In-Depth Information
Steady-state
evaporation
+
q
wy
Steady-state
infiltration
−
q
wy
Cover
Ground
surface
3
Steady-state
flow downward
q
wy
(
−
)
1
2
Steady-state
flow upward
q
wy
(+)
Gravitational
head
Static equilibrium
with water table
(
q
wy
= 0)
y
x
Datum
Water table
z
0
Pore-water pressure head distribution
(
−
)
(+)
Figure 7.22
Hydrostatic equilibrium and steady-state flow conditions in zone of negative pore-
water pressures.
condition at ground surface. The water table acts as the lower
boundary condition and is given a fixed zero pore-water pres-
sure head.
In the laboratory measurement for the coefficient of per-
meability, hydraulic heads are controlled as boundary con-
ditions at the top and bottom of the soil specimen. Let us
consider an unsaturated soil element with one-dimensional
water flow in the
y
-direction (Fig. 7.23). The element has
infinitesimal
dx
,
dy
, and
dz
dimensions. The flow rate
v
wy
is assumed to be positive when water flows upward in the
y
-direction. Continuity requires that the volume of water
flowing in and out of the element be equal for steady-
state conditions:
v
wy
+
where:
v
wy
=
water flow rate across a unit area of the soil
in the
y-
direction and
dx
,
dy
,
dz
=
dimensions in the
x
-,
y
-, and
z
-directions,
respectively.
The net flow through the element can be written as
follows:
d
v
wy
dy
dx dy dz
=
0
(7.28)
Substituting Darcy's law into the above equation yields
d
−
k
wy
u
a
−
u
w
dh
w
/dy
dy
dy
dx dz
d
v
wy
dy
dx dy dz
=
0
(7.29)
−
v
wy
dx dz
=
0
(7.27)
where:
k
wy
(u
a
−
u
w
)
=
water coefficient of permeability, which
varies in the
y
-direction since it is a func-
tion of matric suction,
dh
w
/dy
=
hydraulic head gradient in the
y
-direc-
tion, and
h
w
=
hydraulic head (i.e., gravitational head
plus pore-water pressure head).
Equation 7.29 can be used to solve for the hydraulic head
distribution in the
y
-direction. Since matric suction varies
from one location to another, the coefficient of permeability
also varies. However, for the remainder of the formulation,
Figure 7.23
One-dimensional water flow through an unsaturated
soil element.
Search WWH ::
Custom Search