Porous Media Permeability

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 only for Abaqus/Standard simulations.

The permeability material option:

must be specified for a wetting liquid for an effective stress/wetting liquid diffusion analysis;
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.

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

f (1 + β \sqrt{v_{w} . v_{w}}) = - \frac{k_{s}}{γ_{w}} k . (\frac{\partial u_{w}}{\partial x} - ρ_{w} g),

where

$f = s n v_{w}$ is the volumetric flow rate of wetting liquid per unit area of the porous medium;
$s = \frac{d V_{w}}{d V_{v}}$ is the fluid saturation ( $s = 1$ for a fully saturated medium, $s = 0$ for a completely dry medium);
$n = \frac{d V_{v}}{d V}$ is the porosity of the porous medium;
$e = \frac{d V_{v}}{d V_{g} + d V_{t}}$ is the void ratio;
$d V_{w}$ is the wetting fluid volume of the medium;
$d V_{v}$ is the void volume of the medium;
$d V_{g}$ is the volume of grains of solid material in the medium;
$d V_{t}$ is the volume of trapped wetting liquid in the medium;
$d V$ is the total volume of the medium;
$v_{w}$ is the fluid velocity;
$β (e)$ is a "velocity coefficient," which may depend on the void ratio of the material;
$k_{s} (s)$ is the dependence of permeability on saturation of the wetting liquid such that $k_{s} = 1.0$ at $s = 1.0$ ;
$ρ_{w} = \frac{γ_{w}}{g}$ is the density of the fluid;
$γ_{w}$ is the specific weight of the wetting fluid;
$g$ is the magnitude of the gravitational acceleration;
$k (e, θ, f_{β})$ is the permeability of the fully saturated medium, which can be a function of void ratio ( $e$ ), temperature ( $θ$ ), and/or field variables ( $f_{β}$ );
$x$ is position; and
$g$ is the gravitational acceleration.

Wetting Liquid Specific Weight


Type	Description
Wetting liquid specific weight	Set this parameter equal to the specific weight of the wetting liquid, $γ_{w}$ . The actual specific weight must be given as a nonzero positive value, and the GRAV distributed load type must be used to apply the gravitational loading if a total pressure solution is required

Permeability Types


Type	Description
Isotropic	Specifies a single porous media permeability for all material directions.
Orthotropic	Specifies porous media permeability in three directions.
Anisotropic	Specifies six values of porous media permeability to relate the volumetric fluid flow rates in three directions to the gradients of the effective fluid pressure in three directions.

Isotropic


Input Data	Description
Permeability	Fully saturated isotropic permeability, $k$ .
Void ratio	If the permeability depends on void ratio, enter the value of the void ratio, $e$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Orthotropic


Input Data	Description
k`nn`	Three values to define fully saturated orthotropic permeability: $k 11$ , $k 22$ , and $k 33$ .
Void ratio	If the permeability depends on void ratio, enter the value of the void ratio, $e$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Anisotropic


Input Data	Description
k`nn`	Six values to define fully saturated anisotropic permeability: $k 11$ , $k 12$ , $k 22$ , $k 13$ , $k 23$ , and $k 33$ .
Void ratio	If the permeability depends on void ratio, enter the value of the void ratio, $e$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Permeability Saturation Dependence


Input Data	Description
Permeability factor	Permeability factor, $k_{s}$ that modifies the fully saturated permeability.
Saturation	If the permeability factor depends on saturation, enter the value of the saturation, $s$ . The table must provide $k_{s} = 1.0$ at $s = 1.0$

Permeability Velocity based on Forchheimer's Law


Input Data	Description
Beta	Velocity coefficient, $β$ in Forchheimer's Law.
Void ratio	If the velocity coefficient depends on void ratio, enter the value of the void ratio, $e$ . Only $β > 1.0$ is allowed.