# POROSITY_MODEL

Specifies a porosity model for the flow equation.

## Type

AcuSolve Command

## Syntax

POROSITY_MODEL("name") {parameters...}

## Qualifier

User-given name.

## Parameters

- type (enumerated) [=none]
- Type of the porosity model.
- none
- No porosity.
- constant or const
- Darcy-Forchheimer porosity model. Requires permeability, darcy_coefficient, forchheimer_coefficient and permeability_direction.

- porosity (real) >0 <=1 [=1]
- Porosity value.
- permeability_type (enumerated) [=cartesian]
- Permeability type.
- cartesian
- Use Cartesian coordinate system to specify permeabilities and directions.
- cylindrical
- Use cylindrical coordinate system to specify permeabilities and directions.
- spherical
- Use spherical coordinate system to specify permeabilities and directions.

- direction_1_permeability (real) [=1]
- direction_2_permeability (real) [=1]
- direction_3_permeability (real) [=1]
- Permeability of the porous media in each of the three orthogonal principal axes. Used with constant type. These commands are valid only when permeability_type=cartesian. Also, these commands are equivalent to the previously used permeability array [={1,1,1}], which has been deprecated. AcuSolve will automatically map the permeability_array to the appropriate direction_1_permeability, direction_2_permeability, and direction_3_permeability parameters when reading an input file that uses the deprecated syntax. Note that if both parameters are specified, this mapping will cause the permeability to take precedence over the values that are specified for direction_1_permeability, direction_2_permeability and direction_3_permeability.
- cartesian_permeability_directions (array) [={1,0,0;0,1,0;0,0,1}]
- Orthogonal principal axes of the permeability values. The vectors are ortho-normalized prior to use. Used with constant type. This command is equivalent to the previously used permeability_direction [={1,0,0;0.1.0;0,0,1}]. Note that if both are specified in the input file, the one that appears last will be used for the computation.
- direction_1_permeability_multiplier_function (string) [=none]
- direction_2_permeability_multiplier_function (string) [=none]
- direction_3_permeability_multiplier_function (string) [=none]
- User-given name of the multiplier function for scaling the permeabilities. If none, no scaling is performed. These commands are valid only when permeability_type = cartesian.
- radial_permeability (real) [=1]
- Radial permeability of the porous media in cylindrical and spherical coordinate systems. This command is valid only when permeability_type = cylindrical or spherical.
- axial_permeability (real) [=1]
- Axial permeability of the porous media in cylindrical coordinate system. This command is valid only when permeability_type = cylindrical.
- tangential_permeability (real) [=1]
- Tangential permeability of the porous media in cylindrical and spherical coordinate systems. This command is valid only when permeability_type = cylindrical or spherical.
- cylindrical_permeability_axis (array) [={0,0.0;1,0,0}]
- Principal axis of the axial permeability value. Other two principal axes in cylindrical coordinate are computed internally and ortho-normalized prior to use.
- radial_permeability_multiplier_function (string) [=none]
- User-given name of the multiplier function for scaling the permeabilities. If none, no scaling is performed. This command is valid only when permeability_type = cylindrical or spherical.
- axial_permeability_multiplier_function (string) [=none]
- User-given name of the multiplier function for scaling the permeabilities. If none, no scaling is performed. This command is valid only when permeability_type = cylindrical.
- tangential_permeability_multiplier_function (string) [=none]
- User-given name of the multiplier function for scaling the permeabilities. If none, no scaling is performed. This command is valid only when permeability_type = cylindrical or spherical.
- spherical_permeability_center (array) [={0,0,0}]
- Spherical center of permeability values. This command is valid only when permeability_type = spherical.
- darcy_coefficient (real) [=0]
- Coefficient of Darcy's (linear) term. Used with constant type.
- forchheimer_coefficient (real) [=0]
- Coefficient of Forchheimer's (quadratic) term. Used with constant type.
- darcy_multiplier_function (string) [=none]
- User-given name of the multiplier function for scaling the Darcy coefficient. If none, no scaling is performed.
- forchheimer_multiplier_function (string) [=none]
- User-given name of the multiplier function for scaling the Forchheimer coefficient. If none, no scaling is performed.

## Description

This command specifies a porosity model for the flow (momentum) equations. This model is only applicable to fluid element sets.

```
POROSITY_MODEL( "my porous media" ) {
type = constant
direction_1_permeability = 1.e-6
direction_2_permeability = 1.e-2
direction_3_permeability = 1.e-6
cartesian_permeability_directions = { 1, 0, 0 ;
0, 1, 0 ;
0, 0, 1 ; }
darcy_coefficient = 0
forchheimer_coefficient = 0.5
}
MATERIAL_MODEL( "my material model" ) {
porosity_model = "my porous media"
...
}
ELEMENT_SET( "fluid elements" ) {
material_model = "my material model"
...
}
```

_{1}, u

_{2}, u

_{3}}

^{T}is the velocity vector; f={f

_{1}, f

_{2}, f

_{3}}

^{T}is the porous media contribution, defined in the orthogonal principal axes given below;

**R**is the rotation tensor which rotates

**f**to the global coordinate axes; p is the pressure; is the viscous stress tensor; and

**b**is the specific body-force. The Darcy-Forchheimer porosity model is used with type constant. The porous media forces for this model are given by:

where
$\mu $
is the viscosity, given by the VISCOSITY_MODEL command; the index
i refers to one of the principal orthogonal
directions, given by the permeability_direction array;
C_{Darcy}and
C_{Forch} are, respectively, the
linear and quadratic coefficients of the porosity model, given by
darcy_coefficient and
forchheimer_coefficient; and
K_{i} is the permeability in the
principal direction i , given by permeability.
The porosity for this model is Φ=1.

```
POROSITY_MODEL( "ramped Darcy coefficient" ) {
type = constant
permeability_type = cartesian
direction_1_permeability = 1.e-6
direction_2_permeability = 1.e-2
direction_3_permeability = 1.e-6
cartesian_permeability_directions = { 1, 0, 0 ; 0, 1, 0 ; 0, 0, 1 ; }
darcy_coefficient = 1
forchheimer_coefficient = 0.5
darcy_multiplier_function = "ramped"
}
MULTIPLIER_FUNCTION( "ramped" ) {
type = piecewise_linear
curve_fit_vales = { 1, 0 ; 10, 0.5 }
curve_fit_variable = time_step
}
```

```
POROSITY_MODEL( "ramped Forchheimer coefficient" ) {
type = constant
direction_1_permeability = 1.e-6
direction_2_permeability = 1.e-2
direction_3_permeability = 1.e-6
cartesian_permeability_direction = { 1, 0, 0 ; 0, 1, 0 ; 0, 0, 1 ; }
darcy_coefficient = 0.5
forchheimer_coefficient = 1
forchheimer_multiplier_function = "ramped"
}
```

```
POROSITY_MODEL( "my porous media" ) {
type = constant
permeability_type = cylindrical
radial_permeability = 1.e-6
axial_permeability = 1.e-2
tangential_permeability = 1.e-6
cylindrical_permeability_axis = {0, 0, 0 ; 1, 0, 0 ; }
darcy_coefficient = 1
forchheimer_coefficient = 0.5
}
```

```
POROSITY_MODEL( "my porous media" ) {
type = constant
permeability_type = spherical
radial_permeability = 1.e-2
tangential_permeability = 1.e-6
spherical_permeability_axis = {0, 0, 0 ;
darcy_coefficient = 1
forchheimer_coefficient = 0.5
}
```