Environmental Engineering Reference
In-Depth Information
16.8.1 Partial Differential Equations
for Two-Dimensional Expansive Soil Example
as Uncoupled Analysis
Vector notation will be used for designation of the
x-, y-,
and
z-
directions in order to shorten the lengths of the equations.
The water continuity equation for an unsaturated soil can be
written as follows:
unsaturated soil property functions are required when solv-
ing transient seepage problems: coefficient of water volume
change (or water storage modulus) and coefficient of perme-
ability. The coefficient of water storage and the coefficient
of permeability can be assumed to mainly be functions of
soil suction.
The coefficient of water storage is the arithmetic slope
of the SWCC and is obtained by differentiating the SWCC
with respect to matric suction (i.e.,
m
2
+∇
v
w
=
∂(θ)
∂t
dθ/dψ
). Numer-
ous equations have been proposed to simulate the SWCC
and any one of these equations can be used for determin-
ing the water storage function
m
2
. The Fredlund and Xing
(1994) equation for the SWCC is used to illustrate the appli-
cation of the SWCC to the swelling process:
=
0
(16.78)
where:
∇=
(∂
∂y)j
(∂
/
∂x)i
+
+
(∂
/
∂z)k,
the divergence oper-
ator, and
v
w
j
v
w
i
v
w
k,
Darcy's flux.
v
w
=
+
+
m
f
1
θ
=
θ
s
ln
e
(ψ/a
f
)
n
f
(16.82)
The governing equation for the water phase is obtained
by substituting the time derivative of the water phase con-
stitutive equation and Darcy's law into the water phase
continuity equation 16.78:
+
where:
∂
u
a
−
u
w
k
w
∇
u
w
ρ
w
g
+
Y
ψ
=
soil suction, kPa,
∂ε
v
∂t
β
w
1
+
β
w
2
−∇
=
0
e
=
natural log base, 2.71828,
∂t
θ
s
=
volumetric water content at saturation,
(16.79)
Equation 16.79 can be written as follows for the two-
dimensional case:
θ
=
volumetric water content at any soil suction,
a
f
=
parameter designating the inflection point along the
SWCC, kPa,
∂
u
a
−
u
w
n
f
=
parameter related to the rate of desaturation at the
inflection point on the SWCC, and
k
wx
u
w
ρ
w
g
∂ε
v
∂t
∂
∂x
∂
∂x
β
w
1
+
β
w
2
=
∂t
m
f
=
parameter which is related to the residual water con-
tent of the soil.
k
w
y
u
w
ρ
w
g
+
Y
∂
∂y
∂
∂y
+
(16.80)
The coefficient-of-permeability functions,
k
wx
and
k
wy
can
be estimated using the SWCC and the saturated coefficient
of permeability. There are several empirical equations that
have been proposed for the coefficient of permeability for
an unsaturated soil (e.g., Gardner, 1958b; Fredlund et al.,
1994b; Leong and Rahardjo, 1997a). The permeability
power function proposed by Leong and Rahardjo (1997a)
is used to illustrate the swelling process. The permeability
function for the
y-
direction,
k
wy
, can be written as follows
based on the Fredlund and Xing (1994) equation for the
SWCC:
The analysis of seepage and volume change in an unsatu-
rated, expansive soil requires information on the soil struc-
ture and water phase constitutive surfaces, the coefficient of
permeability function, and the water storage of the soil. The
transient water flow process in a swelling soil is influenced
by the expansiveness of the soil structure (Vu, 2003). The
effects of changes in suction are less when soil suction is
high but increase significantly as suction approaches zero.
The swelling-versus-time formulation can be decoupled
by first solving for changes in matric suction with respect
to time. In this case, no consideration is given to changes in
overall volume and net normal stresses are not allowed to
change during the seepage process. The constitutive surface
for the water phase can be represented by the SWCC and
Eq. 16.80 takes on the following form:
pm
f
1
ln
e
(ψ/a
f
)
nf
k
wy
=
k
sy
(16.83)
+
where:
∂
u
a
−
u
w
k
w
x
u
w
ρ
w
g
∂
∂x
∂
∂x
p
=
estimated parameter based on the analysis of a data
set of measured permeability functions (Leong and
Rahardjo, 1997a; Fredlund et al., 2001) and
m
2
=
∂t
k
w
y
u
w
ρ
w
g
+
Y
∂
∂y
∂
∂y
+
(16.81)
k
sy
=
saturated water coefficient of permeability in the
y-
direction (an equation similar to Eq. 16.83 can also
be written for the
x-
direction using the saturated
coefficient of permeability in the
x-
direction,
k
sx
).
Equation 16.81 applies for both transient and steady-state
seepage conditions in saturated and unsaturated soils. Two
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