Coupled pore fluid diffusion and stress analysis

Some of the more common coupled pore fluid diffusion/stress (and, optionally, thermal) analysis problems that can be analyzed with Abaqus/Standard are:

Saturated flow

Soil mechanics problems generally involve fully saturated flow, since the solid is fully saturated with ground water. Typical examples of saturated flow include consolidation of soils under foundations and excavation of tunnels in saturated soil.

Partially saturated flow

Partially saturated flow occurs when the wetting liquid is absorbed into or exsorbed from the medium by capillary action. Irrigation and hydrology problems generally include partially saturated flow.

Combined flow

Combined fully saturated and partially saturated flow occurs in problems such as seepage of water through an earth dam, where the position of the phreatic surface (the boundary between fully saturated and partially saturated soil) is of interest.

Moisture migration

Although not normally associated with soil mechanics, moisture migration problems can also be solved using the coupled pore fluid diffusion/stress procedure. These problems may involve partially saturated flow in polymeric materials such as paper towels and sponge-like materials; in the biomedical industry they may also involve saturated flow in hydrated soft tissues.

Combined heat transfer and pore fluid flow

In some applications, such as a source of heat buried in soil, it is important to model the coupling between the mechanical deformation, pore fluid flow, and heat transfer. In such problems the difference in the thermal expansion coefficients between the soil and the pore fluid often plays an important role in determining the rate of diffusion of the pore fluid and heat from the source.

A porous medium is modeled in Abaqus/Standard by a conventional approach that considers the medium as a multiphase material and adopts an effective stress principle to describe its behavior. The porous medium modeling provided considers the presence of two fluids in the medium. One is the “wetting liquid,” which is assumed to be relatively (but not entirely) incompressible. Often the other is a gas, which is relatively compressible. An example of such a system is soil containing ground water. When the medium is partially saturated, both fluids exist at a point; when it is fully saturated, the voids are completely filled with the wetting liquid. The elementary volume, $dV$, is made up of a volume of grains of solid material, $d{V}_g$; a volume of voids, $d{V}_v$; and a volume of wetting liquid, $d{V}_w\le d{V}_v$, that is free to move through the medium if driven. In some systems (for example, systems containing particles that absorb the wetting liquid and swell in the process) there may also be a significant volume of trapped wetting liquid, $d{V}_t$.

The porous medium is modeled by attaching the finite element mesh to the solid phase; fluid can flow through this mesh. The mechanical part of the model is based on the effective stress principle defined in Effective stress principle for porous media.

The model also uses a continuity equation for the mass of wetting fluid in a unit volume of the medium. This equation is described in Continuity statement for the wetting liquid phase in a porous medium. It is written with pore pressure (the average pressure in the wetting fluid at a point in the porous medium) as the basic variable (degree of freedom 8 at the nodes). The conjugate flux variable is the volumetric flow rate at the node, $\mathbf{f}$. The porous medium is partially saturated when the pore liquid pressure, $u_w$, is negative.

Optionally, heat transfer due to conduction in the soil skeleton and pore fluid, as well as convection in the pore fluid, can also be modeled. This capability represents an enhancement to the basic pore fluid flow capabilities discussed in the earlier paragraphs and requires the use of coupled temperature–pore pressure elements that have temperature as an additional degree of freedom (degree of freedom 11 at the nodes) in addition to the pore pressure and the displacement components. When you use the coupled temperature–pore pressure elements, Abaqus solves the heat transfer equation in addition to and in a fully coupled manner with the continuity equation and the mechanical equilibrium equations. Only linear brick, first-order axisymmetric, and second-order modified tetrahedrons are available for modeling coupled heat transfer with pore fluid flow and mechanical deformation.

The coupled pore fluid diffusion/stress analysis capability can provide solutions either in terms of total or “excess” pore fluid pressure. The excess pore fluid pressure at a point is the pore fluid pressure in excess of the hydrostatic pressure required to support the weight of pore fluid above the elevation of the material point. The difference between total and excess pore pressure is relevant only for cases in which gravitational loading is important; for example, when the loading provided by the hydrostatic pressure in the pore fluid is large or when effects like “wicking” (transient capillary suction of liquid into a dry column) are being studied. Total pore pressure solutions are provided when the gravity distributed load is used to define the gravity load on the model. Excess pore pressure solutions are provided in all other cases; for example, when gravity loading is defined with body force distributed loads.

Steady-state coupled pore pressure/effective stress analysis assumes that there are no transient effects in the wetting liquid continuity equation; that is, the steady-state solution corresponds to constant wetting liquid velocities and constant volume of wetting liquid per unit volume in the continuum. Thus, for example, thermal expansion of the liquid phase has no effect on the steady-state solution: it is a transient effect. Therefore, the time scale chosen during steady-state analysis is relevant only to rate effects in the constitutive model used for the porous medium (excluding creep and viscoelasticity, which are disabled in steady-state analysis).

Mechanical loads and boundary conditions can be changed gradually over the step by referring to an amplitude curve to accommodate possible geometric nonlinearities in the response.

The steady-state coupled equations are strongly unsymmetric; therefore, the unsymmetric matrix solution and storage scheme is used automatically for steady-state analysis steps (see Defining an analysis).

If heat transfer is modeled using the coupled temperature–pore pressure elements, the steady-state solution neglects all transient effects in the heat transfer equation and provides only the steady-state temperature distribution.

SOILS

Step module: Create Step: General: Soils: Basic: Pore fluid response: Steady state

In a transient coupled pore pressure/effective stress analysis the backward difference operator is used to integrate the continuity equation and the heat transfer equation (if heat transfer is modeled): this operator provides unconditional stability so that the only concern with respect to time integration is accuracy. You can provide the time increments, or they can be selected automatically.

The coupled partially saturated flow equations are strongly unsymmetric, so the unsymmetric solver is used automatically if you request partially saturated analysis (by including absorption/exsorption behavior in the material definition). The unsymmetric solver is also activated automatically when gravity distributed loading is used during a soils consolidation analysis.

For fully saturated flow analyses in which finite-sliding coupled pore pressure-displacement contact is modeled using contact pairs, certain contributions to the model's stiffness matrix are unsymmetric. Using the unsymmetric solver can sometimes improve convergence in such cases since Abaqus does not automatically do so.

For fully saturated flow analyses in which heat transfer is also modeled, the contributions to the model's stiffness matrix arising from convective heat transfer due to pore fluid flow are unsymmetric. Using the unsymmetric solver can sometimes improve convergence in such cases since Abaqus does not automatically do so.

When coupled temperature–pore pressure elements are used, heat transfer is modeled in these elements by default. However, you may optionally choose to switch off heat transfer within these elements during some steps in the analysis. This feature may be helpful in reducing computation time during certain phases in the analysis when heat transfer is not an important part of the overall physics of the problem.

SOILS, CONSOLIDATION, HEAT=NO

In coupled problems where two or more different fields are being solved, you must be careful when choosing the units of the problem. If the choice of units is such that the numbers generated by the equations for the different fields differ by many orders of magnitude, the precision on some computers may be insufficient to resolve the numerical ill-conditioning of the coupled equations. Therefore, choose units that avoid badly conditioned matrices. For example, consider using units of Mpascal instead of pascal for the stress equilibrium equations to reduce the disparity between the magnitudes of the stress equilibrium equations and the pore flow continuity equations.