Environmental Engineering Reference
In-Depth Information
coefficient of transmission are assumed to be negligible.
Therefore, Eqs. 16.17 and 16.36 are used as the PDEs for the
water and air phases, respectively. The transient water flow
equation (i.e., Eq. 16.17) can be written in the following
finite difference form:
equation into Eq. 16.38 gives:
u
a(i,j
+
1
)
−
u
a(i,j)
u
a(i,
1
+
j)
−
u
a(i,j)
=
C
a
C
w
t
t
u
w
(i
+
1
,j)
−
2
u
w
(i,j)
+
u
w
(i
−
1
,j)
c
v
C
a
−
y
2
u
a(i
+
1
,j)
−
2
u
a(i,j)
+
u
a(i
−
1
,j)
u
w
(i,j
+
1
)
−
u
w
(i,j)
u
a(i,j
+
1
)
−
u
a(i,j)
c
v
+
(16.41)
=−
C
w
y
2
t
t
u
w
(i
+
1
,j)
−
2
u
w
(i,j)
+
u
w
(i
−
1
,j)
Simplifying and rearranging Eq. 16.41 allows
the
c
v
+
y
2
unknown pore-air pressure (i.e., at a given time step
j
1)
to be put on the left-hand side and all known variables from
the previous time step (i.e.,
j
) to be put on the right-hand
side of the equation:
+
(16.37)
where:
β
w
g
1
i
=
space increment in the
y
-direction and
β
a
f
1
C
a
u
a(i,j
+
1
)
=
u
a(i,j)
−
+
j
=
time increment.
C
a
C
w
(16.42)
The pore-water pressure and pore-air pressure at a given
time step are computed from the known values at the pre-
vious time step using Eqs. 16.40 and 16.42, respectively.
It is possible to march forward to the next time step when
computations for the pore-water pressure and pore-air pres-
sure are complete for all depth steps. The above procedure
can be repeated until equilibrium has been achieved for the
air and water phases. The correct converged solution of the
finite difference equations are obtained when
β
w
and
β
a
are
maintained less than 0.5 (Desai and Christian, 1977).
The initial conditions in the soil mass and the bound-
ary conditions for the problem must be established prior
to performing the finite difference computations. The initial
conditions for one-dimensional consolidation constitute the
initial pore pressures before load is applied and the excess
pore pressures corresponding to the instant after the appli-
cation of the total load. The excess pore-air and pore-water
pressures are calculated using the pore-air and pore-water
pressure parameters.
The condition on all boundaries of a problem must be
specified during the consolidation process. The boundary
pore pressures must be set to the initial values at all times for
free-draining conditions. The pore pressure gradients must
be set to zero at an impervious boundary since there is no
flow across the boundary.
A mixed boundary condition can also be specified for
special applications that might occur in laboratory tests. As
an example, laboratory equipment is commonly designed
such that air flows upward while water flows downward dur-
ing a test. In this case, the top boundary is a free-draining
boundary with respect to the air phase but has an imper-
vious boundary with respect to the water phase. Reverse
conditions exist at the bottom boundary.
The dissipation of the excess pore-air and pore-water pres-
sures can be computed using the finite difference schemes
shown by Eqs. 16.40 and 16.42, respectively, once the ini-
tial and final boundary conditions are specified. The cal-
culated pore pressure changes can be used to compute the
−
C
a
C
w
−
1
1
The transient air flow equation (i.e., Eq. 16.36) can be
written in the following finite difference form:
u
a(i,j
+
1
)
−
u
a(i,j)
u
w
(i,j
+
1
)
−
u
w
(i,j)
=−
C
a
t
t
u
a(i
+
1
,j)
−
2
u
a(i,j)
+
u
a(i
−
1
,j)
c
v
+
y
2
(16.38)
The pore-water pressure is solved by multiplying Eq.
16.38 by
−
C
w
and substituting the resulting equation into
Eq. 16.37:
u
w
(i,j
+
1
)
−
u
w
(i,j)
u
w
(i,
1
+
j)
−
u
w
(i,j)
=
C
a
C
w
t
t
u
a(i
+
j,
1
)
−
2
u
a(i,j)
+
u
a(i
−
1
,j)
c
v
C
w
−
y
2
u
w
(i
+
1
,j)
−
2
u
w
(i,j)
+
u
w
(i
−
1
,j)
c
v
+
y
2
(16.39)
Simplifying and rearranging Eq. 16.39 allows the
unknown pore-water pressure (i.e., at a given time step,
j
+
1) to be put on the left-hand side of the equation:
β
a
f
1
(16.40)
β
w
g
1
C
w
u
w
(i,j
+
1
)
=
u
w
(i,j)
+
C
a
C
w
−
1
−
1
−
C
a
C
w
where:
t
y
2
c
v
β
w
=
t
y
2
c
v
β
a
=
g
1
=
u
w
(i
+
1
,j)
−
2
u
w
(i,j)
+
u
w
(i
−
1
,j)
f
1
=
u
a(i
+
1
,j)
−
2
u
a(i,j)
+
u
a(i
−
1
,j)
Similarly, the pore-air pressure can be obtained by mul-
tiplying Eq. 16.37 by
−
C
a
, and substituting the resulting
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