Product: ABAQUS/Standard
The purpose of this example is to validate the ABAQUS capability to solve coupled fluid flow problems in partially saturated porous media where the effects of gravity are important. For this purpose 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.
The column of material is 1 m high and 0.1 m wide. We model the problem with 10 CPE8RP plane strain elements. In addition, input files containing element types CPE4P, CPE4RP, CPE6MPH, CAX4PH, CAX6MP, C3D8P, and C3D10MP are included for verification purposes. The mesh is shown in Figure 1.8.4–1. We constrain all horizontal displacements (the flow problem is one-dimensional). In the deforming column problem we constrain the vertical displacements at the bottom of the column, while in the rigid column problem we constrain all the vertical displacements.
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 1.8.4–2, 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,” Section 6.7.2 of the ABAQUS Analysis User's Manual. 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.
The weight is applied by GRAV loading. In the case of the deforming column an initial step of *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 *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. UTOL, 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,” Section 1.8.1. For the parameters of this problem the initial time increment is chosen as 20 seconds.
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 1.8.4–3. Figure 1.8.4–4 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 1.8.4–5) 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 1.8.4–6 for both the deformable and rigid columns: as expected, more fluid is lost in the deforming column case. Figure 1.8.4–7 and Figure 1.8.4–8 show the time history of fluid saturation and pore pressure at six points along the height of the column.
Element type C3D8P (deforming column).
Element type C3D10MP (deforming column).
Element type CAX4PH (deforming column).
Element type CAX6MP (deforming column).
Element type CPE4P (deforming column).
Rigid column (element type CPE4RP).
Rigid column (element type CPE6MPH).
Deforming column (element type CPE8RP).
Rigid column (element type CPE8RP).
Rigid column simulated by declaring CPE8RP elements as rigid.
Liakopoulos, A. C., Transient Flow Through Unsaturated Porous Media, D. Eng. dissertation, University of California, Berkeley, 1965.
Schrefler, B. A., and L. Simoni, “A Unified Approach to the Analysis of Saturated-Unsaturated Elastoplastic Porous Media,” Numerical Methods in Geomechanics, vol. 1, pp. 205–212, 1988.
