Environmental Engineering Reference
In-Depth Information
Rearranging Eqs. 7.55 and 7.56 yields the expressions for
the rate of water flow in the
x
- and
y
-directions:
performed on an arithmetic scale and gives rise to the water
storage modulus
m
2
.
The volume-mass constitutive equations required for
derivation of the transient water flow equation is presented in
Chapter 13 on volume change (i.e., stress-deformation). The
net flux of water can be computed from the time derivative of
the water phase constitutive equation. The time derivatives of
the total stress and the pore-air pressure are equal to zero since
both pressures are assumed to remain constant with time.
The continuity condition can be satisfied by equating the
divergence of the water flow rates (
∂
v
x
/∂x
k
wxx
∂h
w
∂x
∂h
w
∂y
v
wx
=−
+
k
wxy
(7.57)
k
wyx
∂h
w
∂x
∂h
w
∂y
v
wy
=−
+
k
wyy
(7.58)
where:
∂
v
y
/∂y
)tothe
time derivative of the constitutive equation for the water
phase:
∂
∂x
+
k
w
1
cos
2
α
k
w
2
sin
2
α
k
wxx
=
+
k
wxy
=
k
w
1
−
k
w
2
sin
α
cos
α
k
wyx
=
k
w
1
−
k
w
2
sin
α
cos
α
k
wxx
k
w
1
sin
2
α
∂h
w
∂x
∂h
w
∂y
k
w
2
cos
2
α
k
wyy
=
+
+
k
wxy
k
wyx
Equations 7.57 and 7.58 provide the flow rates in the
x
-
and
y
-directions in terms of the major and minor coefficients
of permeability. These flow rate expressions can then be
used in the formulation for unsteady-state seepage analyses.
∂
∂y
∂h
w
∂x
∂h
w
∂y
m
2
ρ
w
g
∂h
w
∂t
+
+
k
wyy
=
(7.60)
Equation 7.60 is the governing partial differential equation
for unsteady-state water seepage in an anisotropic soil. The
pore-air pressure is assumed to remain constant with time.
In many cases, the directions for the
major and minor
coeffi-
cients of permeability coincide with the
x
- and
y
-directions,
respectively. In this case, the
α
angle is equal to zero, and
the governing equation can be simplified by setting the
α
angle to zero:
∂
∂x
7.5.3 Water Phase Partial Differential Equation
for Transient Flow
The water phase partial differential equation for transient
two-dimensional seepage can be derived by considering the
continuity for the water phase. The assumption is made that
there are no changes in total stress with time in this analysis.
The net flux of water through an element of unsaturated soil
(Fig. 7.24) can be computed from the volume rates of water
entering and leaving the element over a period of time:
∂
V
w
/V
0
∂t
k
w
1
k
w
2
∂h
w
∂x
∂
∂y
∂h
w
∂y
m
2
ρ
w
g
∂h
w
∂t
+
=
(7.61)
The
k
w
1
and
k
w
2
terms in Eq. 7.60 are the major and
minor coefficients of permeability in the
x
- and
y
-directions,
respectively. These permeability coefficients are a function
of matric suction that can vary with location in the
x
- and
∂
v
wy
∂y
∂
v
wx
∂x
=
+
(7.59)
y
-directions [i.e.,
k
w
1
u
a
−
u
w
]. For
isotropic
soil conditions, the
k
w
1
and
k
w
2
terms are equal
[i.e.,
k
w
1
=
u
w
and
k
w
2
u
a
−
where:
u
w
], and the transient seepage
equation can be further simplified as follows:
k
w
u
a
−
k
w
2
=
V
w
=
volume of water in the element,
V
0
=
initial overall volume of the element (i.e.,
dx
,
dy
,
dz
),
k
w
k
w
∂
∂x
∂h
w
∂x
∂
∂y
∂h
w
∂y
m
2
ρ
w
g
∂h
w
∂t
+
=
(7.62)
dx
,
dy
,
dz
=
dimensions
in
the
x
-,
y
-,
and
z
-directions, respectively, and
∂(V
w
/V
0
)/∂t
=
rate of change in the volume of water in
the soil element with respect to the initial
volume of the element.
Rearranging the above equation gives
∂
2
h
w
∂x
2
∂
2
h
w
∂y
2
∂k
w
∂x
∂h
w
∂x
∂k
w
∂y
∂h
w
∂y
m
2
ρ
w
g
∂h
w
∂t
(7.63)
k
w
+
+
k
w
+
=
The difference between a steady-state and unsteady-state
water flow formulation lies in the representation of changes
in the amount of water in the soil,
V
w
, with respect to time.
The amount of water in the soil for any applied suction
is represented by the SWCC. The SWCC must be written
in terms of the volumetric water content when calculating
water storage. Any mathematical representation of the (vol-
umetric) SWCC can be differentiated to yield the change in
volumetric water content,
θ
, of the soil with respect to soil
u
w
/ρ
w
g
for the hydraulic head
h
w
and
rearranging the equation result in an equation written in
terms of pore-water pressure:
Substituting
y
+
∂
2
u
w
∂x
2
c
v
k
w
∂
2
u
w
∂y
2
c
v
k
w
∂k
w
∂x
∂u
w
∂x
∂k
w
∂y
∂u
w
∂y
c
v
c
v
+
+
+
∂k
w
∂y
∂u
w
∂t
+
c
g
=
(7.64)
suction [i.e.,
dθ/d
u
a
−
u
w
]. The differentiation must be
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