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.
 
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