Porous media such as soils, rocks, bones, and soft tissue are solid skeletons that contain pores connected and filled with fluids. Porous media analysis includes both the deformation of solid skeletons and the flow of fluids, and they are coupled. In general, the diffusion of fluid pressure is accompanied by the consolidation of the porous media. The analysis can include heat transfer within the porous media. The process is also time-dependent, and generally a transient analysis. The analysis can include pore-pressure contact between parts. (See the contact elements for pore-pressure contact options.)

Both fully and partially saturated porous media can be modeled. Soil embankments and dams are typical examples of partially saturated porous media. These structures are often characterized by a phreatic line. The soil below the phreatic line is fully saturated, while the soil above is partially saturated. Partially saturated flow results in negative pore pressures. The presence of negative pore pressures is beneficial to the soil; it assures some cohesion in the soil, essential for maintaining its structural integrity.

The Biot consolidation theory, upon which a pore fluid diffusion and structural analysis is based, has many applications in civil, petroleum, nuclear, and biomedical engineering, including:

Soil is a typical porous media type consisting of aggregates of mineral particles which, together with air and/or water in the void spaces, form a three-phase system. When all voids are filled with water, the medium is fully saturated. When the voids are partially filled with water, the medium is partially saturated. Soil mechanics is an engineering discipline dealing with the engineering properties (stress and strain behavior) and applications of soil. The Mechanical APDL program provides a soil-analysis type (ANTYPE,SOIL) for modeling soil in which the porous media can be fully saturated (a two-phase system of solid and fluid), or partially saturated (a three-phase system of solid, fluid, and gas).

The porous media (soil) analysis is based on extended Biot theory that considers the medium a multiphase material and adopts an effective stress principle to describe its behavior. The solid part of the model is represented by the effective stress. Fluid flow is based on a continuity equation for the mass of fluid in a unit volume of the medium. Optionally, heat transfer can be considered due to thermal conduction in the solid medium.

Command input:

! define soil analysis 
ANTYPE,SOIL

The constitutive behavior of the solid skeleton is based on the effective stress principal describing the mechanical response of material.

The material models supported in the coupled structural-pore-fluid-diffusion analysis include elasticity (isotropic, orthotropic and anisotropic), porous elasticity, Mohr Coulomb plasticity, Cam-clay plasticity, jointed rock plasticity, extended Drucker-Prager and Drucker-Prager-based concrete plasticity.

For more information about the supported material models and how to define them, see:

! define permeability table
TB,PM,mat,,,PERM
! define Biot constants
TB,PM,mat,,,BIOT
! define solid properties
TB,PM,mat,,,SP
! define fluid properties
TB,PM,mat,,,FP
! define degree of saturation table
TB,PM,mat,,,DSAT
! define relative permeability table
TB,PM,mat,,,RPER
! define gravity magnitude
TB,PM,mat,,,GRAV

! define solid-skeleton thermal expansion
TB,CTE,mat,,,
! define fluid thermal expansion
TB,CTE,mat,,,FLUID 

! define thermal-conductivity table
TB,PM,mat,,,COND
! define solid-skeleton specific-heat capacity 
TB,PM,mat,,,SPHT
! define fluid specific-heat capacity
TB,PM,mat,,,FLSPHT 

Porous media (soil) analysis (ANTYPE,SOIL) is based on the extended Biot consolidation theory that considers the medium as a multiphase material and adopts an effective stress principle to describe its behavior. The solid part of the model is represented by the effective stress. Fluid flow is based on a continuity equation for the mass of fluid in a unit volume of the medium.

A porous medium model considers the presence of a single fluid phase in the medium (fully saturated flow), or a mixture of fluid and air in the medium (partially saturated flow). The basic governing law for the flow of fluids through porous media is Darcy's law, formulated by the French civil engineer Henry Darcy in 1856 based on experiments involving vertical water filtration through sand beds:

where:

= flow flux

= relative permeability

= permeability

= degree of fluid saturation

= specific weight of fluid

= pore pressure

= gravity load direction

For more information, see Porous Media Mechanics in the Material Reference.

The default load direction is -y axis (0,-1,0) in the global coordinate system.

In the soil solution, you can account for the pore-fluid weight by defining fluid properties (TB,PM), then using the soil-solution option to set the specific-weight load direction and apply the fluid properties (SSOPT,SFSW,gvx,gvy,gvz,,ON).

Command input:

! define fluid properties
TB,PM,mat,,,FP
! define soil solution type
ANTYP,SOIL
! set specific-weight direction, and apply pore-fluid weight (Par5 = ON)
SSOPT,SFSW,gvx,gvy,gvz,,ON

See Porous Media Mechanics in the Material Reference.

Soil stresses are caused by applied loads and the soil's own weight. Soil under ground is already stressed due to its weight even before any applied loads.

Soil stress caused by its weight is often called the geostatic stress state. The program's geostatic stress equilibrium can balance the initial stress, soil weight, and initial pore pressure under the applied loads and boundary conditions. It is usually the first step in soil analysis and is followed by a soil-consolidation analysis with applied loading. The stress state can be complex.

A simple example occurs when the soil ground surface is horizontal and the nature of the soil varies very little along the horizontal direction (for example, sedimentary soil). In this case, no shear stress exists, and the vertical stress at a given depth can then be calculated from the weight of the soil above the specified depth.

Assume that the unit weight of soil is constant through the depth. The vertical stress is then:

where is the depth and is the unit weight of soil. In this case, the vertical stress varies linearly with the depth.

If the unit weight of the soil changes with the depth (as usually the soil becomes denser with the depth because of the higher compression stress due to the increasing weight of the soil), the vertical stress can then be calculated by integrating the unit weight over the depth as:

You can use the initial state capability to define depth-dependent initial stress, etc.

The geostatic stress equilibrium is part of the soil solution. To initiate a geostatic solution, specify a soil analysis type (ANTYPE,SOIL) and set the soil solution option to geostatic (SSOPT,GEOSTATIC). Set the number of substeps to one (NSUBST,1).

Command input:

! set analysis type to soil
ANTYPE,SOIL
! set solution option to geostatic
SSOPT,GEOSTATIC

Automatic time-stepping is enabled by default. You can specify initial, minimum, and maximum time increments (DELTIM or NSUBST).

Automatic time-stepping uses an internal heuristic to adjust the time increment. To adjust the accuracy of time integration of the flow continuity equation, you can add an additional time-stepping control (CUTCONTROL,DPPLMT) to control the maximum incremental pore pressure in a time step allowed. The program restricts the time increment to ensure that the specified value is not exceeded at all active pore-pressure degrees of freedom (excluding degrees of freedom defined by boundary condition).

The nature of the governing equations and implementation scheme is such that a relationship exists between the minimum time step and element size. If the time step is too small, you may observe spurious oscillations in the solution, leading to inaccurate results and convergence problems. If you observe oscillations, try using a finer mesh or a larger time step.

Command input:

! auto time stepping (default)
AUTOTS,ON
! define initial, min, and max time increments
DELTIM,0.01,0.0001,1
! define max incremental pore pressure allowed in a step
CUTCONTROL,DPPLIMIT,1.5

Soil analysis is generally transient, ending when the solution reaches the specified time period or steady state.

By default, the solution ends when the specified time period has been completed. However you can set the solution to end when a steady-state condition is reached or when the time period ends, whichever comes first.

The steady-state condition is reached when the maximum incremental pore pressure in a step is less than the user-specified reference value. Set a soil analysis solution option to define the steady-state pore-pressure increment (SSOPT).

Command input:

! define steady-state incremental pore-pressure value in a step
SSOPT,STOP,SSTATE,1.0

You can apply an initial condition of displacements and pore pressure at nodes (IC).

Initial stress, strain, pore pressure, void volume ratio, degree of saturation, and relative permeability can be applied to elements or element nodes via the initial state capability.

Specifying an initial stress state results in an initial effective stress in the porous media (that is, the initial stress is directly applied as an initial effective stress).

Use initial degree of saturation and relative permeability with the degree of saturation (TB,PM,,,,DSAT) and relative permeability (TB,PM,,,,RPER) tables, respectively, as another way of defining initial pore pressure. For a particular initial-state of degree of saturation or relative permeability, the corresponding pore pressure value is retrieved from the relevant table. If both the initial-state degree of saturation and relative permeability are defined, initial-state degree of saturation takes precedence. If initial-state pore pressure is also defined with initial state degree of saturation (or relative permeability), the pore-pressure value is overwritten with the pore pressure corresponding to the initial-state degree of saturation value (or initial state relative permeability value).

Most geomechanical analyses start with a geostatic stress step to establish the equilibrium stress state of the soils under geostatic loading, ensuring that the subsequent consolidation analysis begins from an equilibrium state.

Initial stress, pore pressure, and void volume ratio are subject to soil weight and are therefore depth-dependent. Initial pore pressure can be defined at element integration points or nodes. The initial void volume ratio (ratio of the void volume to the solid skeleton volume) can be defined at element integration points or nodes; the evolution of the void volume depends on the deformation of the mechanical response of different phases of material.

You can use the function-based initial state (INISTATE) state capability to define depth-dependent initial stress, depth-dependent initial pore pressure, and depth-dependent initial void volume ratio. With function-based initial-state, initial stress, pore pressure and void volume ratio are linear functions of coordinates; the data field can vary only linearly with a defined coordinate in a user-specified coordinate system.

Command input:

INISTATE,SET,DATA,FUNC
INISTATE,SET,DTYP,S
! define initial stress SXX as a linear function of Y
INISTATE,DEFINE,ALL,,,,LINY,AXX,BXX  

The commands define initial stress as a linear function of coordinate Y:

You can specify the same variation for all six stress components.

For more information, see Initial State in the Advanced Analysis Guide.

In a soil analysis, you can specify both predefined field variables (such as temperature) and user-defined field variables to define the field-dependent material properties. For more information defining field-dependent material properties, see TBFIELD. For more information about defining field variables, see Initial State in the Advanced Analysis Guide and INISTATE.

For the temperature field variable, define uniformly distributed temperature (TUNIF). You can also define nodal (BF) and element (BFE) temperatures.

For user-defined field variables, specify the field values (INISTATE).

Command input:

TUNIF,temp
BF,node,TEMP,v1,v2, ...
BFE,elem,TEMP,1,v1,v2,v4,v5
INISTATE,DEFINE

Coupled structural pore-pressure elements include displacements (UX, UY, UZ) and pore pressure (PRES) as degrees of freedom. When heat transfer is included in the analysis, the temperature is a degree of freedom as well. The displacement, pore-pressure, and temperature boundary conditions can be applied directly to the nodes (D).

If the temperature is applied to the nodes, the heat flow around the nodes is assumed to maintain the temperature applied.

If the pore pressure is applied to the nodes, the free fluid flow around the nodes is assumed to maintain the pressure applied.

Command input:

! prescribed displacement boundary UX to the node
D,node,UX,value
! prescribed pore pressure boundary to the node 
D,node,PRES,value
! prescribed temperature boundary to the node 
D,node,TEMP,value

 

2.12.11.2. Specific Weight of Fluid and Porous Media Bulk

Define the specific weight of fluid and solid via the appropriate property option (TBOPT on TB,PM). To account for the specific weight of fluid or solid (or both), define the specific weight load direction , then apply the loads (SSOPT,SFSW).

To account for the specific weight of fluid in the fluid flow, define it via the fluid-property option (TBOPT = FP).

To account for the specific weight of porous-media bulk gravity load, define both the solid property (TBOPT = SP) and the fluid property (TBOPT = FP).

The specific weight of bulk is calculated as: where:  = specific weight of solid  = specific weight of fluid  = degree of saturation of fluid  = porosity

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Important:  Specific weight is always applied at the beginning of a solution as step loading; therefore, do not define the specific weight of bulk and the gravity load (ACEL) at the same time for the same model.

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Command input:

! apply bulk specific weight
SSOPT,SFSW,gx,gy,gz,ON,OFF
! apply fluid specific weight
SSOPT,SFSW,gx,gy,gz,OFF,ON
! apply both fluid and bulk specific weight
SSOPT,SFSW,gx,gy,gz,ON,ON

The coupled pore fluid diffusion, thermal, and structural analysis is available for plane strain, axisymmetric, and three-dimensional problems with the coupled structure pore-pressure-thermal elements. In addition to the displacement degrees of freedom, the elements have both pore-pressure degrees of freedom and temperature degrees of freedom at the corner nodes. The following table describes the coupled pore-pressure-thermal element types within the context of a steady-state and full transient analysis:

The solution results output includes nodes and elements.

Node output includes standard mechanical solution outputs such as displacement and reaction forces. The pore-pressure and temperature degrees of freedom are also available for postprocessing. The following variables and records are available for node postprocessing:

Element outputs include standard mechanical solution variables such as (total) stress, effective stress, strain, plastic strain (if any), and strain energy densities. The pore-fluid-related quantities are fluid flux, pore pressure and void volume ratio. The temperature-related quantities are thermal flux and temperature. The following variables and records are available for element postprocessing:

Following is the general process for performing a coupled structural-pore-fluid-diffusion analysis: