For this example we compare
Abaqus
results with the experimental work of Liakopoulos (1965). Most experiments of
this type are performed to check the hydraulic parameters; and, therefore, some
assumptions about the mechanical behavior have to be made in the numerical
model. Schrefler and Simoni (1988) have provided a numerical solution for the
Liakopoulos experiment, and this example follows their assumptions about the
mechanical behavior.
The Liakopoulos experiment consists of the drainage of water from a vertical
column of sand. A column of perspex, 1 m high, is filled with Del Monte sand
and instrumented to measure the moisture pressure at various points along the
height of the column. Prior to the start of the experiment water is added
continually at the top of the column and allowed to drain freely at the bottom
of the column. The flow is regulated until zero pore pressure readings are
obtained throughout the column. At this point flow is stopped and the
experiment starts: the top of the column is made impermeable and the water is
allowed to drain out of the column, under gravity. Pore pressure profiles in
the column are measured during the drainage transient.
We investigate two cases: one in which the column is not allowed to deform
(uncoupled flow problem), and the other in which we consider the deformation of
the sand (coupled problem). The latter is expected to be a closer
representation of the physical experiment.
Material
The properties used in this example pertaining to the partially saturated
flow behavior of the material are taken from Liakopoulos (1965) and are as used
by Schrefler and Simoni (1988): the pore pressure/saturation relationship is
shown in
Figure 2,
and the permeability of the fully saturated material is 4.5 × 10−6
m/sec. The partially saturated permeability decreases linearly from this value
to a value of 3.0 × 10−6 m/sec at a saturation of 0.85 and remains
constant below that. A bulk modulus of 2 GPa is used for the water. The
mechanical properties for the sand are not given by Liakopoulos. Following
Schrefler and Simoni (1988), we assume the material is elastic with Young's
modulus 1.3 MPa and Poisson's ratio 0. We also assume that the mass density of
the dry material is 1500 kg/m3, which is typical of sand.
The initial void ratio of the material is 0.4235. The initial conditions for
pore pressure and saturation correspond to the fully saturated state of the
sand at the beginning of the experiment: the initial saturation is 1.0, and the
initial pore pressure is 0.0. There is some steady-state flow under these
initial conditions because the zero gradient in pore pressure does not
equilibrate the specific weight of the fluid.
In the deforming column case the initial conditions for effective stress are
calculated from the density of the dry material and fluid, the initial
saturation and void ratio, and the initial pore pressures using equilibrium
considerations and the effective stress principle. The procedure used is
detailed in
Geostatic Stress State.
It is important to specify the correct initial conditions for this type of
problem; otherwise, the system may be so far out of equilibrium initially that
it may fail to start because converged solutions cannot be found.
Loading and controls
The weight is applied by gravity loading. In the case of the deforming
column an initial step of a geostatic analysis is performed to establish the
initial equilibrium state; the initial conditions in the column exactly balance
the weight of the fluid and dry material so that no deformation takes place,
while the zero pore pressure boundary conditions enforce the initial
steady-state of fluid flow. Then the fluid is allowed to drain through the
bottom of the column by prescribing zero pore pressures at these nodes during a
transient soils consolidation step. The fluid will drain until the pressure
gradient is equal to the weight of the fluid, at which time equilibrium is
established.
The transient analysis is performed using automatic time incrementation. The
pore pressure tolerance that controls the automatic incrementation is set to a
large value since we expect the nonlinearity of the material to restrict the
size of the time increments during the transient stages of the analysis and we
do not wish to impose any further control on the accuracy of the time
integration.
The choice of initial time step in these transient partially saturated flow
problems is important. This is discussed in
Partially saturated flow in a porous medium.
For the parameters of this problem the initial time increment is chosen as 20
seconds.
Results and discussion
The profiles of pore pressure obtained in the coupled analysis (deformable
column case) at different times during the drainage process are compared to the
experimental results in
Figure 3.
Figure 4
shows the corresponding comparison for the uncoupled analysis (rigid column).
The results of the coupled analysis are closer to the experiment than those of
the uncoupled analysis; in particular, the uncoupled analysis tends to
overestimate the pore pressures in the early stages of the transient. As the
transient continues, the material deformation slows (see the displacement
histories of six points along the height of the column in
Figure 5)
and, therefore, the rigid column assumption becomes closer to reality; as
steady-state is approached, both numerical solutions are in good agreement with
the experiment. At steady state, the pore pressure gradient is equal to the
weight density of the fluid, as required by Darcy's law. The time histories of
the volume of fluid lost through the bottom of the column are shown in
Figure 6
for both the deformable and rigid columns: as expected, more fluid is lost in
the deforming column case.
Figure 7
and
Figure 8
show the time history of fluid saturation and pore pressure at six points along
the height of the column.
