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 commands are referenced by MATERIAL_MODEL commands, which in turn are referenced by ELEMENT_SET commands:

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"
   ...
}

The POROSITY_MODEL command modifies the momentum equation as follows:(1)

where

ρ

is the density; Φ is the porosity; u={u₁, u₂, u₃}^Tis the velocity vector; f={f₁, f₂, f₃}^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:(2)

where $μ$ 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_Darcyand 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.

The darcy_multiplier_function parameter may be used to uniformly scale the Darcy coefficient. The value of this parameter refers to the user-given name of a MULTIPLIER_FUNCTION command in the input file. For example, a Darcy coefficient ramped from zero to 0.5 over the first 10 time steps may be specified by:

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
}

Similarly, forchheimer_multiplier_function may be used to uniformly scale the Forchheimer coefficient. The value of this parameter refers to the user-given name of a MULTIPLIER_FUNCTION command in the input file. For example, using the same MULTIPLIER_FUNCTION command as above, a Forchheimer coefficient ramped from zero to 0.5 over the first 10 time steps may be specified by:

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"
}

While the default coordinate system is Cartesian, you can also use cylindrical or spherical coordinate systems to specify permeabilities and their directions. The POROSITY_MODEL commands below are an example of cylindrical coordinate system:

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
}

and for spherical coordinate system:

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
}