The coupled pore-pressure thermal elements used in analyses involving porous media are listed in Coupled Pore-Pressure-Thermal Element Support.
The program models porous media containing fluid by treating the porous media as a multiphase material and applying an extended version of Biot's consolidation theory. The flow is considered to be a single-phase fluid. The porous media can be fully or partially saturated.[[426]][[432]] Optionally, heat transfer in the porous media can also be considered.
Following are the governing equations for Biot consolidation problems with heat transfer:
 | (10–113) |
where:
| σ | = | Total Cauchy stress |
|
| = |
 = matrix differentiation operator (3D form
shown) |
| = | Bulk density of porous media |
| = | Displacement |
| = | Bulk specific weight of porous media |
| = | Gravity load direction (not to be confused with gravity magnitude) |
| = | Flow flux vector |
| = |
 = Gradient operator (3D form shown) |
| = | Biot coefficient |
|
| = | Volumetric strain of the solid skeleton |
| = | Pore pressure |
| = | Compressibility parameter |
| = | Degree of saturation of fluid |
| = | Free strain |
| = | Temperature |
| = | Density-specific heat term 
|
| = | Porosity |
| = | Density of solid skeleton and fluid |
| = | Specific heats of solid skeleton and fluid |
| = | Thermal conductivity |
The total stress relates to the effective stress 
and pore pressure by:
where:
 = Second-order identity tensor |
The relationship between the effective stress and the elastic strain of solid skeletons is given by:
where:
 = Second-order elastic strain tensor |
 = Fourth-order elasticity tensor |
The relationship between the fluid flow flux and the pore pressure is described by Darcy's Law:
where:
 = second-order permeability tensor |
 = relative permeability |
 = specific weight of fluid |
For displacement 
, pressure 
, and temperature 
as the unknown degrees of freedom, linearizing the governing equations
gives:
 | (10–114) |
The matrices are:
|
|
|
|
|
|
|
|
|
|
|
|
|
where:
 = domain |
 = strain-displacement operator matrix |
 = displacement interpolation |
 = pressure interpolation |
 = temperature interpolation |
 = thermal load vector |
The load force vector 
includes the body force and surface traction boundary conditions, the
vector 
includes the flow source, and the vector 
includes the heat source. ([426])
Combining the linearized equations for porous media with the equation of motion gives the matrix equation:
 | (10–115) |
where:
 = structural damping matrix |
The structural damping matrix can be input as Rayleigh damping (TB,SDAMP,,,,ALPD and/or TB,SDAMP,,,,BETD).
Additional Information
For related information, see the following documentation: