Environmental Engineering Reference
In-Depth Information
problem at hand should be solved in a one-, two-, or three-
dimensional context.
Second, the geotechnical engineer must be satisfied that an
appropriate partial differential equation can be written for a
representative elemental volume of the continuum. The par-
tial differential equation must adequately describe the physi-
cal processes that control air flow through a porous media.
Third, it must be possible to measure or estimate the air
phase soil properties and other soil properties relevant to the
problem being solved.
Fourth, boundary conditions must be assessed for all
boundaries. Initial conditions also need to be established
when modeling time-dependent flow problems. Transient
air flow problems can be highly nonlinear and the computed
heads can be dependent upon the selected initial values.
The modeler should also assess whether a steady-state or
transient analysis is required. It is generally advisable to
first perform a steady-state analysis.
Fifth, the solutions should be performed and the results
should be studied from the standpoint of being reasonable.
There are many potential sources of error when undertaking
numerical modeling studies and it is important that “checks”
be performed to ensure that the computer output is correct.
Figure 9.34 Pore-air pressure distribution during measurement of
coefficient of diffusion D .
pressures at each end of the specimen (i.e., u a =
u ab =
0 . 0
at y
L ) are the boundary con-
ditions. Substituting the boundary conditions into the dif-
ferential equation yields a linear equation for the diffus-
ing pore-air pressure distribution in the y -direction [i.e.,
u a =
=
0 . 0 and u a =
u at at y
=
(y/L) u atm ].
9.10.4 Two-Dimensional Air Flow Well (or Trench)
Example
A two-dimensional cross section is passed through a 2-m-
wide trench that was dug to a depth of 20 m. The lower
6 meters of the trench was sealed off and subjected to a pres-
sure of 20 kPa (gauge). Steady-state air flow conditions were
established from the bottom of the trench to the surrounding
atmosphere.
Relevant coordinates of the geometry of the problem are
shown in Fig. 9.35. The soil is fine sand with an air coefficient
of permeability of 2 . 18
9.10.2 Types of Boundary Conditions for Air
Flow Problems
The first air flow problem illustrates the diffusion of air
through a saturated ceramic disk. The analysis is one dimen-
sional and the initial air pressures are known. It is not
necessary to use numerical modeling software to solve the
first problem.
Other example problems are more complex and require
numerical modeling software in order to obtain a solution.
Air fluxes near ground surface may be part of the input
information that needs to be provided by the modeler. Often
it is the air pressure isochrones that are of greatest interest.
10 4 m/s. The right and left sides
of the model were assumed to be impervious. The ground
surface and the top portion of the trench were exposed to
atmospheric pressure (i.e., 101.0 kPa absolute). The objective
was to calculate the dissipation of pore-air pressures from the
bottom of the trench to the surrounding atmosphere.
Figure 9.36 shows the generated finite element mesh along
with the contour of pore-air pressure under steady-state air
flow using the SVAir software. The highest pore-air pressure
gradients are near the packers that were placed at a distance
of 6m above the base of the trench. The highest pore-air
pressure gradient locations are also the locations where the
finite element mesh needs to be refined (i.e., smaller ele-
ments) in order to ensure a correct numerical solution. Also
shown are the air pressure contours throughout the sand.
×
9.10.3 Example 1: One-Dimensional Air Flow
Examples
The diffusion of air through a saturated ceramic high-air-
entry disk is an example of steady-state air diffusion through
water. Another example is the diffusion of air through a sat-
urated soil specimen. In each case, the diffused air pressure
is dissipated across a region of water.
The measurement of the coefficient of diffusion can be
used as an example of steady-state air diffusion through
water. The coefficient is assumed to be a constant. The
partial differential equation describing air diffusion takes
the same form as that for air flow through an unsaturated
soil (i.e., d 2 u a / dy 2
9.10.5 Two-Dimensional Air Flow Well
with Clay Layer
The above example involving the trench in sand was reana-
lyzed with one change made to the geometry of the problem,
as shown in Fig. 9.37. In this case, a 2-m-thick layer of
fine silt was shown to be located 2m below the bottom of
0). The pore-air pressure distribution
through the soil specimen can be assumed to be linear.
An example showing the pore-air and pore-water pressure
distributions across a soil specimen during the measurement
of the coefficient of diffusivity is shown in Fig. 9.34. The air
=
 
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