Environmental Engineering Reference
In-Depth Information
7.3 DARCY'S LAW FOR UNSATURATED SOILS
The flow of water in a saturated soil can be described using
Darcy's law. Darcy (1856) postulated that the rate of water
flow through a soil mass was proportional to the hydraulic
head gradient and could be described using the following
equation:
∂h
w
∂y
v
w
=−
k
w
(7.9)
where:
v
w
=
flow rate of water,
k
w
=
coefficient of permeability with respect to the
water phase, and
Figure 7.6
Experimental verification of Darcy's law for water
flow through an unsaturated soil (after Childs and Collis-George,
1950).
∂h
w
/
∂y
=
hydraulic head gradient in the
y
-direction des-
ignated as
i
wy
.
The coefficient of proportionality between the flow rate of
water and the hydraulic head gradient is called the coeffi-
cient of permeability,
k
w
. The coefficient of permeability is
relatively constant for a specific saturated soil. Darcy's law
can also be written for the
x
- and
z
-directions. The negative
sign in the flow equation indicates that water flows in the
direction of a decreasing hydraulic head.
Darcy's law also applies for the flow of water through an
unsaturated soil (Buckingham, 1907; Richards, 1931; Childs
and Collis-George, 1950). However, the coefficient of per-
meability in an unsaturated soil cannot generally be assumed
to be constant. Rather, the coefficient of permeability is a
variable which is predominantly a function of the water con-
tent or the matric suction of the unsaturated soil.
Water can be visualized as flowing only through the pore
spaces that are filled with water. Air-filled pores are non-
conductive channels to the flow of water. Therefore, the
air-filled pores in an unsaturated soil can be considered
as behaving similarly to the solid phase, and the soil can
be treated as a saturated soil having reduced water content
(Childs, 1969). The validity of Darcy's law can be veri-
fied for an unsaturated soil using an experiment similar to
that used for the verification of the flow law for saturated
soils. However, the volume of water (or water content) must
remain constant while the hydraulic head gradient is varied.
Experiments to verify Darcy's law for unsaturated soils
have been performed, and key experimental results are pre-
sented in Fig. 7.6 (Childs and Collis-George, 1950). A col-
umn of unsaturated soil held at a uniform water content
and a constant-water-pressure head was subjected to vari-
ous gradients of gravitational head. The results indicate that
at a specific water content the coefficient of permeability
k
w
is constant for various hydraulic head gradients applied to
the unsaturated soil. In other words, the rate of water flow
through an unsaturated soil is linearly proportional to the
hydraulic head gradient, with the coefficient of permeability
being a constant. This behavior is similar to that observed
for saturated soils.
The results of the Childs and Collis-George (1950) experi-
ment confirm that Darcy's law can also be applied to unsatu-
rated soils. However, it needs to be noted that the magnitude
of the coefficient of permeability will differ at various vol-
umetric water contents
θ
in an unsaturated soil.
7.3.1 Coefficient of Permeability with Respect to
Water Phase
The coefficient of permeability with respect to the water
phase,
k
w
, is a measure of the ease with which water can
flow in the space available for water flow in the unsaturated
soil. The coefficient of permeability depends upon the prop-
erties of the fluid and the properties of the porous medium.
Different types of fluid (e.g., water and oil) or different types
of soil (e.g., sand and clay) produce different values for the
coefficient of permeability
k
w
.
7.3.2 Fluid and Porous Medium Components
The coefficients of permeability with respect to water,
k
w
,
can be expressed in terms of the intrinsic permeability
K,
which can mathematically be described as follows:
ρ
w
g
ν
γ
ν
K
k
w
=
K
=
(7.10)
where:
intrinsic permeability of the soil, m
2
,
K
=
gravitational acceleration, m/s
2
,
g
=
ν
=
dynamic viscosity, cP,
density, kg/m
3
, and
ρ
=
unit weight of the medium, N/m
3
.
γ
=
Fluids properties can be “scaled” to the properties of water
at standard conditions (Parker et al., 1987). The intrinsic
permeability is approximately 10
−
12
m
2
for a soil that has
an assumed hydraulic conductivity of 10
−
5
m/s. The density
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