Permeability

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

f (1 + β \sqrt{v_{w} \cdot v_{w}}) = - \frac{k_{s}}{γ_{w}} k \cdot (\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 (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);
$n = \frac{d V_{v}}{d V}$: 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 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;
$d V$: is the total volume of the medium;
$v_{w}$: is the fluid velocity;
$β (e)$: is a “velocity coefficient,” which may be dependent 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 $s = 1.0$ ;
$ρ_{w} = γ_{w} / g$: is the density of the fluid;
$γ_{w}$: is the specific weight of the wetting liquid;
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, common in soil consolidation problems), temperature ( $θ$ ), and/or field variables ( $f_{β}$ );
$u_{w}$: is the wetting liquid pore pressure;
$x$: is position; and
$g$: 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

\bar{k} = \frac{k_{s}}{(1 + β \sqrt{v_{w} \cdot v_{w}})} k,

so that Forchheimer's law can also be written as

f = - \frac{\bar{k}}{γ_{w}} \cdot (\frac{\partial u_{w}}{\partial x} - ρ_{w} g) .

The fully saturated permeability, $k$ , is typically obtained from experiments under low fluid velocity conditions. $k$ can be defined as a function of void ratio, e, (common in soil consolidation problems) and/or temperature, $θ$ . The void ratio can be derived from the porosity, n, using the relationship $e = n / (1 - n)$ . 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, $\bar{k}$ (units of ${LT}^{- 1}$ ), as the “hydraulic conductivity” of the porous medium and define the permeability as

\hat{K} = \frac{ν}{g} \frac{k_{s}}{(1 + β \sqrt{v_{w} \cdot v_{w}})} k = \frac{ν}{g} \bar{k},

where $ν$ 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 $\hat{K}$ has dimensions $L^{2}$ (or Darcy). If the permeability is available in this form, it must be converted such that the appropriate values of $k$ are used in Abaqus/Standard.

Specifying the Permeability

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.

Isotropic Permeability

For isotropic permeability in Abaqus/Standard define one value of the fully saturated permeability at each value of the void ratio.

Orthotropic Permeability

For orthotropic permeability in Abaqus/Standard define three values of the fully saturated permeability ( $k_{11}$ , $k_{22}$ , and $k_{33}$ ) at each value of the void ratio.

Anisotropic Permeability

For fully anisotropic permeability in Abaqus/Standard define six values of the fully saturated permeability ( $k_{11}$ , $k_{12}$ , $k_{22}$ , $k_{13}$ , $k_{23}$ , and $k_{33}$ ) at each value of the void ratio.

Velocity Coefficient

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

Saturation Dependence

In Abaqus/Standard you can define the dependence of permeability, $\bar{k}$ , 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 $s \geq 1.0$ . The tabular definition of $k_{s} (s)$ must specify $k_{s} = 1.0$ for $s \geq 1.0$ .

Specific Weight of the Wetting Liquid

In Abaqus/Standard the specific weight of the fluid, $γ_{w}$ , 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).

Specifying a Stabilization Coefficient

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.

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).