26.6.2 Permeability

Products: Abaqus/Standard  Abaqus/CFD  Abaqus/CAE  

Overview

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. It can be specified in Abaqus/Standard and Abaqus/CFD.

Permeability in Abaqus/Standard:

  • must be specified for a wetting liquid for an effective stress/wetting liquid diffusion analysis (see Coupled pore fluid diffusion and stress analysis, Section 6.8.1);

  • 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.

Permeability in Abaqus/CFD:

Permeability in Abaqus/Standard

Permeability is defined for pore fluid flow.

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, e, (common in soil consolidation problems) and/or temperature, . The void ratio can be derived from the porosity, n, using the relationship . Up to six variables may be needed to define the fully saturated permeability, depending on whether isotropic, orthotropic, or fully anisotropic permeability is to be modeled (discussed below).

Alternative definition of permeability

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), g is the magnitude of the gravitational acceleration, and has dimensions (or Darcy). If the permeability is available in this form, it must be converted such that the appropriate values of are used in Abaqus/Standard.

Specifying the permeability

Permeability in Abaqus/Standard 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.

Isotropic permeability

For isotropic permeability in Abaqus/Standard 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: OtherPore FluidPermeability: Type: Isotropic


Orthotropic permeability

For orthotropic permeability in Abaqus/Standard 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: OtherPore FluidPermeability: Type: Orthotropic


Anisotropic permeability

For fully anisotropic permeability in Abaqus/Standard 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: OtherPore FluidPermeability: 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: OtherPore FluidPermeability: SuboptionsVelocity Dependence


Saturation dependence

In Abaqus/Standard 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: OtherPore FluidPermeability: SuboptionsSaturation Dependence


Specific weight of the wetting liquid

In Abaqus/Standard 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=type, SPECIFIC=

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: OtherPore FluidPermeability: Specific weight of wetting liquid:


Permeability in Abaqus/CFD

For flows in fluid-saturated porous medium, the momentum equation in its simplest form can be written as

where the first term on the right-hand side is the Darcy drag and the second term is the inertial drag (also called form drag or Forchheimer drag). In the above equation

is the intrinsic average of the pressure (average taken over the fluid-phase only);

is the extrinsic or superficial velocity vector, where the average is taken over a representative volume incorporating both the solid (matrix) and the fluid phases;

is the density of the fluid;

is the viscosity of the fluid;

is the permeability of the porous medium (units of L2); and

is the dimensionless inertial or form drag coefficient and, in general, is a function of the porosity .

The inertial drag coefficient, , is usually a function of the porosity . In Abaqus/CFD the Ergun's relation is used, which is given by

where the constant is set by default as .

A widely used model to specify the permeability as a function of porosity is the Carman-Kozeny relation, which is given by

where represents the Carman-Kozeny constant (parameter that is geometry dependent) and represents the average radius of the porous particles/fibers.

Specifying the permeability

Permeability in Abaqus/CFD can be isotropic (with dependence only on porosity) or specified using a Carman-Kozeny relation.

Isotropic permeability

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

Input File Usage:          
*PERMEABILITY, TYPE=ISOTROPIC

Abaqus/CAE Usage:   

Property module: material editor: OtherPore FluidPermeability: Type: Isotropic (CFD)


Carman-Kozeny model

For the Carman-Kozeny relation, you can define the permeability by specifying , the Carman-Kozeny constant, and , the average pore-particle/fiber radius.

Input File Usage:          
*PERMEABILITY, TYPE=CARMAN KOZENY

Abaqus/CAE Usage:   

Property module: material editor: OtherPore FluidPermeability: Type: Carman-Kozeny


Inertial drag coefficient

The value of the constant in the expression for the inertial drag coefficient, , can be set to any user-specified value. By default, the value of is 0.142887.

Input File Usage:          
*PERMEABILITY, TYPE=type, INERTIAL DRAG COEFFICIENT=

Abaqus/CAE Usage:   

Property module: material editor: OtherPore FluidPermeability: Inertial drag coefficient:


Elements

In Abaqus/Standard permeability can be used only in elements that allow for pore pressure (see Choosing the appropriate element for an analysis type, Section 27.1.3). Permeability can be used with any fluid element in Abaqus/CFD.

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26.6.2 Permeability

Products: Abaqus/Standard  Abaqus/CFD  Abaqus/CAE  

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Overview

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. It can be specified in Abaqus/Standard and Abaqus/CFD.

Permeability in Abaqus/Standard:

  • must be specified for a wetting liquid for an effective stress/wetting liquid diffusion analysis (see Coupled pore fluid diffusion and stress analysis, Section 6.8.1);

  • 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.

Permeability in Abaqus/CFD:

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Permeability in Abaqus/Standard

Permeability is defined for pore fluid flow.

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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.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

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, e, (common in soil consolidation problems) and/or temperature, . The void ratio can be derived from the porosity, n, using the relationship . Up to six variables may be needed to define the fully saturated permeability, depending on whether isotropic, orthotropic, or fully anisotropic permeability is to be modeled (discussed below).

Your query was poorly formed. Please make corrections.
Alternative definition of permeability

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), g is the magnitude of the gravitational acceleration, and has dimensions (or Darcy). If the permeability is available in this form, it must be converted such that the appropriate values of are used in Abaqus/Standard.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Specifying the permeability

Permeability in Abaqus/Standard 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.

Your query was poorly formed. Please make corrections.
Isotropic permeability

For isotropic permeability in Abaqus/Standard 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: OtherPore FluidPermeability: Type: Isotropic


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Orthotropic permeability

For orthotropic permeability in Abaqus/Standard 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: OtherPore FluidPermeability: Type: Orthotropic


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Anisotropic permeability

For fully anisotropic permeability in Abaqus/Standard 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: OtherPore FluidPermeability: Type: Anisotropic


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

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: OtherPore FluidPermeability: SuboptionsVelocity Dependence


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Saturation dependence

In Abaqus/Standard 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: OtherPore FluidPermeability: SuboptionsSaturation Dependence


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Specific weight of the wetting liquid

In Abaqus/Standard 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=type, SPECIFIC=

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: OtherPore FluidPermeability: Specific weight of wetting liquid:


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Permeability in Abaqus/CFD

For flows in fluid-saturated porous medium, the momentum equation in its simplest form can be written as

where the first term on the right-hand side is the Darcy drag and the second term is the inertial drag (also called form drag or Forchheimer drag). In the above equation

is the intrinsic average of the pressure (average taken over the fluid-phase only);

is the extrinsic or superficial velocity vector, where the average is taken over a representative volume incorporating both the solid (matrix) and the fluid phases;

is the density of the fluid;

is the viscosity of the fluid;

is the permeability of the porous medium (units of L2); and

is the dimensionless inertial or form drag coefficient and, in general, is a function of the porosity .

The inertial drag coefficient, , is usually a function of the porosity . In Abaqus/CFD the Ergun's relation is used, which is given by

where the constant is set by default as .

A widely used model to specify the permeability as a function of porosity is the Carman-Kozeny relation, which is given by

where represents the Carman-Kozeny constant (parameter that is geometry dependent) and represents the average radius of the porous particles/fibers.

Your query was poorly formed. Please make corrections.

Specifying the permeability

Permeability in Abaqus/CFD can be isotropic (with dependence only on porosity) or specified using a Carman-Kozeny relation.

Your query was poorly formed. Please make corrections.
Isotropic permeability

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

Input File Usage:          
*PERMEABILITY, TYPE=ISOTROPIC

Abaqus/CAE Usage:   

Property module: material editor: OtherPore FluidPermeability: Type: Isotropic (CFD)


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Carman-Kozeny model

For the Carman-Kozeny relation, you can define the permeability by specifying , the Carman-Kozeny constant, and , the average pore-particle/fiber radius.

Input File Usage:          
*PERMEABILITY, TYPE=CARMAN KOZENY

Abaqus/CAE Usage:   

Property module: material editor: OtherPore FluidPermeability: Type: Carman-Kozeny


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Inertial drag coefficient

The value of the constant in the expression for the inertial drag coefficient, , can be set to any user-specified value. By default, the value of is 0.142887.

Input File Usage:          
*PERMEABILITY, TYPE=type, INERTIAL DRAG COEFFICIENT=

Abaqus/CAE Usage:   

Property module: material editor: OtherPore FluidPermeability: Inertial drag coefficient:


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Elements

In Abaqus/Standard permeability can be used only in elements that allow for pore pressure (see Choosing the appropriate element for an analysis type, Section 27.1.3). Permeability can be used with any fluid element in Abaqus/CFD.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.