Permeability

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

$\mathbf{f}=sn{\mathbf{v}}_w$

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

$s=\frac{d{V}_w}{d{V}_v}$

is the fluid saturation ($s=1$ for a fully saturated medium, $s=0$ for a completely dry medium);

$n=\frac{d{V}_v}{dV}$

is the porosity of the porous medium;

$e=\frac{d{V}_v}{\left(d{V}_g+d{V}_t\right)}$

is the void ratio;

$d{V}_w$

is the wetting fluid volume in the medium;

$d{V}_v$

is the void volume in 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;

$dV$

is the total volume of the medium;

${\mathbf{v}}_w$

is the fluid velocity;

$\beta \left(e\right)$

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

$k_s\left(s\right)$

is the dependence of permeability on saturation of the wetting liquid such that $k_s=1.0$ at $s=1.0$;

${\rho }_w={\gamma }_w/g$

is the density of the fluid;

${\gamma }_w$

is the specific weight of the wetting liquid;

g

is the magnitude of the gravitational acceleration;

$\mathbf{k}\left(e,\theta ,f_{\beta }\right)$

is the permeability of the fully saturated medium, which can be a function of void ratio (e, common in soil consolidation problems), temperature ($\theta$), and/or field variables ($f_{\beta }$);

$u_w$

is the wetting liquid pore pressure;

$\mathbf{x}$

is position; and

$\mathbf{g}$

is the gravitational acceleration.

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, $\mathbf{k}$, is typically obtained from experiments under low fluid velocity conditions. $\mathbf{k}$ can be defined as a function of void ratio, e, (common in soil consolidation problems) and/or temperature, $\theta$. The void ratio can be derived from the porosity, n, using the relationship $e=n/\left(1-n\right)$. 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).

Permeability in Abaqus/Standard can be isotropic, orthotropic, or fully anisotropic. For nonisotropic permeability a local orientation (see Orientations) must be used to specify the material directions.

PERMEABILITY, TYPE=ISOTROPIC

Property module: material editor: OtherPore FluidPermeability: Type: Isotropic

PERMEABILITY, TYPE=ORTHOTROPIC

Property module: material editor: OtherPore FluidPermeability: Type: Orthotropic

PERMEABILITY, TYPE=ANISOTROPIC

Property module: material editor: OtherPore FluidPermeability: Type: Anisotropic

Abaqus/Standard assumes that $\beta =0.0$ by default, meaning that Darcy's law is used. If Forchheimer's law is required ($\beta >0.0$), $\beta \left(e\right)$ must be defined in tabular form.

PERMEABILITY, TYPE=VELOCITY

Property module: material editor: OtherPore FluidPermeability: SuboptionsVelocity Dependence

In Abaqus/Standard you can define the dependence of permeability, $\overline{\mathbf{k}}$, on saturation, s, by specifying $k_s$. Abaqus/Standard assumes by default that $k_s=s^3$ for $s<1.0$; $k_s=1.0$ for $s\ge 1.0$. The tabular definition of $k_s\left(s\right)$ must specify $k_s=1.0$ for $s\ge 1.0$.

PERMEABILITY, TYPE=SATURATION

Property module: material editor: OtherPore FluidPermeability: SuboptionsSaturation Dependence

In Abaqus/Standard the specific weight of the fluid, ${\gamma }_w$ , must be specified correctly even if the analysis does not consider the weight of the wetting liquid (that is, if excess pore fluid pressure is calculated).

PERMEABILITY, TYPE=type, SPECIFIC=${\gamma }_w$

Property module: material editor: OtherPore FluidPermeability: Specific weight of wetting liquid: ${\gamma }_w$

In Abaqus/Standard spurious pressure oscillations can occur in the solution for materials with very low permeability. You can add a stabilization term to the element operator and the right-hand-side term by projecting the pressure into the strain space to eliminate the pressure oscillations. For more information, see Spurious Oscillations due to Small Time Increments.

PERMEABILITY, STABILIZE=stabilization_coeff_value