20.7.2 Permeability

**Products: **ABAQUS/Standard ABAQUS/CAE

Permeability:

is the relationship between the volumetric flow rate per unit area of a particular wetting liquid through a porous medium and the gradient of the effective fluid pressure;

must be specified for a wetting liquid in ABAQUS/Standard for an effective stress/wetting liquid diffusion analysis (see “Coupled pore fluid diffusion and stress analysis,” Section 6.7.1);

is defined, in general, by Forchheimer's law, which accounts for changes in permeability as a function of fluid flow velocity; and

can be isotropic, orthotropic, or fully anisotropic and can be given as a function of void ratio, saturation, temperature, and field variables.

Forchheimer's law

According to Forchheimer's law, high flow velocities have the effect of reducing the effective permeability and, therefore, “choking” pore fluid flow. As the fluid flow velocity reduces, Forchheimer's law approximates the well-known Darcy's law. Darcy's law can, therefore, be used directly in ABAQUS/Standard by omitting the velocity-dependent term in Forchheimer's law.

Forchheimer's law is written as

where

is the volumetric flow rate of wetting liquid per unit area of the porous medium (the effective velocity of the wetting liquid);

is the fluid saturation ( for a fully saturated medium, for a completely dry medium);

is the porosity of the porous medium;

is the void ratio;

is the wetting fluid volume in the medium;

is the void volume in the medium;

is the volume of grains of solid material in the medium;

is the volume of trapped wetting liquid in the medium;

is the total volume of the medium;

is the fluid velocity;

is a “velocity coefficient,” which may be dependent on the void ratio of the material;

is the dependence of permeability on saturation of the wetting liquid such that at ;

is the density of the fluid;

is the specific weight of the wetting liquid;

*g*

is the magnitude of the gravitational acceleration;

is the permeability of the fully saturated medium, which can be a function of void ratio (*e*, common in soil consolidation problems), temperature (), and/or field variables ();

is the wetting liquid pore pressure;

is position; and

is the gravitational acceleration.

Permeability definitions

Permeability can be defined in different ways by different authors; caution should, therefore, be used to ensure that the specified input data are consistent with the definitions used in ABAQUS/Standard.

Permeability in ABAQUS/Standard is defined as

so that Forchheimer's law can also be written as

The fully saturated permeability, , is typically obtained from experiments under low fluid velocity conditions. can be defined as a function of void ratio,

Some authors refer to the definition of permeability used in ABAQUS/Standard, (units of ), as the “hydraulic conductivity” of the porous medium and define the permeability as

where is the kinematic viscosity of the wetting liquid (the ratio of the liquid's dynamic viscosity to its mass density),

Specifying the permeability

Permeability can be isotropic, orthotropic, or fully anisotropic. For non-isotropic permeability a local orientation (see “Orientations,” Section 2.2.5) must be used to specify the material directions.

For isotropic permeability define one value of the fully saturated permeability at each value of the void ratio.

Input File Usage: | *PERMEABILITY, TYPE=ISOTROPIC |

ABAQUS/CAE Usage: | Property module: material editor: Type: Isotropic |

For orthotropic permeability define three values of the fully saturated permeability (, , and ) at each value of the void ratio.

Input File Usage: | *PERMEABILITY, TYPE=ORTHOTROPIC |

ABAQUS/CAE Usage: | Property module: material editor: Type: Orthotropic |

For fully anisotropic permeability define six values of the fully saturated permeability (, , , , , and ) at each value of the void ratio.

Input File Usage: | *PERMEABILITY, TYPE=ANISOTROPIC |

ABAQUS/CAE Usage: | Property module: material editor: Type: Anisotropic |

Velocity coefficient

ABAQUS/Standard assumes that by default, meaning that Darcy's law is used. If Forchheimer's law is required (), must be defined in tabular form.

Input File Usage: | *PERMEABILITY, TYPE=VELOCITY |

This must be a repeated use of the *PERMEABILITY option for the same material, since must also be defined. |

ABAQUS/CAE Usage: | Property module: material editor: SuboptionsVelocity Dependence |

Saturation dependence

You can define the dependence of permeability, , on saturation, *s*, by specifying . ABAQUS/Standard assumes by default that for ; for . The tabular definition of must specify for .

Input File Usage: | *PERMEABILITY, TYPE=SATURATION |

This must be a repeated use of the *PERMEABILITY option for the same material, since must also be defined. |

ABAQUS/CAE Usage: | Property module: material editor: SuboptionsSaturation Dependence |

Specific weight of the wetting liquid

The specific weight of the fluid, , must be specified correctly even if the analysis does not consider the weight of the wetting liquid (i.e., if excess pore fluid pressure is calculated).

Input File Usage: | *PERMEABILITY, TYPE= |

The SPECIFIC parameter must be defined in conjunction with the fully saturated *PERMEABILITY option for a given medium. |

ABAQUS/CAE Usage: | Property module: material editor: Specific weight of wetting liquid: |

Permeability can be used only in elements that allow for pore pressure (see “Choosing the appropriate element for an analysis type,” Section 21.1.3).