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For flows in fluid-saturated porous medium, the momentum equation in its simplest form can be written as

\nabla p = - \frac {\mu}{K} \mathbf {v} - \frac {\rho c _ {F}}{K ^ {\frac {1}{2}}} | \mathbf {v} | \mathbf {v},

where the first term on the right-hand side is the Darcy drag and the second term is the inertial drag (also called form drag or Forchheimer drag). In the above equation

p	is the intrinsic average of the pressure (average taken over the fluid-phase only);
v	is the extrinsic or superficial velocity vector, where the average is taken over a representative volume incorporating both the solid (matrix) and the fluid phases;
ρ	is the density of the fluid;
μ	is the viscosity of the fluid;
K	is the permeability of the porous medium (units of $L^{2}$ ); and
$c_{F}$	is the dimensionless inertial or form drag coefficient and, in general, is a function of the porosity $\epsilon$ .

The inertial drag coefficient, c _ { F } , , is usually a function of the porosity . In Abaqus/CFD the Ergun’s relation is used, which is given by

c _ {F} = \frac {C}{\sqrt {\epsilon^ {3}}},

where the constant is set by default as \begin{array} { r } { \frac { 1 . 7 5 } { \sqrt { 1 5 0 } } = 0 . 1 4 2 8 8 7 . } \end{array}

A widely used model to specify the permeability as a function of porosity is the Carman-Kozeny relation, which is given by

K = \frac {r _ {f} ^ {2}}{4 k _ {k c}} \frac {\epsilon^ {3}}{(1 - \epsilon) ^ {2}},

where k _ { k c } represents the Carman-Kozeny constant (parameter that is geometry dependent) and r _ { f } represents the average radius of the porous particles/fibers.

Specifying the permeability

Permeability in Abaqus/CFD can be isotropic (with dependence only on porosity) or specified using a Carman-Kozeny relation.

Isotropic permeability

For isotropic permeability define one value of the fully saturated permeability at each value of the porosity.

\begin{array} { r l r } { \mathrm { l n p u t \ F i l e \ U s a g e : } \quad } & { { } \quad } & { \ast \mathrm { P E R M E A B I L I T Y , \ T Y P E = I S O T R O P I C } } \end{array}

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Type: Isotropic (CFD)

Carman-Kozeny model

For the Carman-Kozeny relation, you can define the permeability by specifying k _ { k c } . , the Carman-Kozeny constant, and r _ { f } , the average pore-particle/fiber radius.

Input File Usage: *PERMEABILITY, TYPE=CARMAN KOZENY

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Type: Carman-Kozeny

Inertial drag coefficient

The value of the constant in the expression for the inertial drag coefficient, c _ { F } , can be set to any user-specified value. By default, the value of is 0.142887.

Input File Usage: *PERMEABILITY, TYPE=type, INERTIAL DRAG COEFFICIENT=

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Permeability: Inertial drag coefficient:

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,” Section 27.1.3). Permeability can be used with any fluid element in Abaqus/CFD.

26.6.3 POROUS BULK MODULI

Products: Abaqus/Standard Abaqus/CAE

References

• “Pore fluid flow properties,” Section 26.6.1
• “Material library: overview,” Section 21.1.1
• *POROUS BULK MODULI
• “Defining porous bulk moduli” in “Defining a fluid-filled porous material,” Section 12.12.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

Overview

The porous bulk moduli:

• must be defined whenever the compressibility of the solid grains or the compressibility of the permeating fluid is to be considered in the analysis of a porous medium; and
• must be defined when a swelling gel is modeled (“Moisture swelling,” Section 26.6.6).

Defining porous bulk moduli

You can specify the bulk modulus of the solid grains and the bulk modulus of the fluid as functions of temperature. If either modulus is omitted or set to zero, that phase of the material is assumed to be fully incompressible.

Input File Usage: *POROUS BULK MODULI

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Porous Bulk Moduli

Elements

The porous bulk moduli can be defined only for elements that allow for pore pressure (see “Choosing the appropriate element for an analysis type,” Section 27.1.3).

26.6.4 SORPTION

Products: Abaqus/Standard Abaqus/CAE

References

• “Pore fluid flow properties,” Section 26.6.1
• “Material library: overview,” Section 21.1.1
• *SORPTION
• “Defining sorption” in “Defining a fluid-filled porous material,” Section 12.12.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

Overview

Sorption:

• defines a porous material’s absorption/exsorption behavior under partially saturated flow conditions; and
• is used in the analysis of coupled wetting liquid flow and porous medium stress (“Coupled pore fluid diffusion and stress analysis,” Section 6.8.1).

Sorption

A porous medium becomes partially saturated when the total pore liquid pressure, u _ { w } , becomes negative (see “Effective stress principle for porous media,” Section 2.8.1 of the Abaqus Theory Guide). Negative values of u _ { w } represent capillary effects in the medium. For u _ { w } < 0 it is known that the saturation lies within certain limits that depend on the value of the capillary pressure, - u _ { w } (see “Continuity statement for the wetting liquid phase in a porous medium,” Section 2.8.4 of the Abaqus Theory Guide). Typical forms of these limits are shown in Figure 26.6.4–1. We write these limits as s ^ { a } \leq s \leq s ^ { e } , where s ^ { a } ( u _ { w } ) is the limit at which absorption will occur (so that \dot { s } > 0 ) , and s ^ { e } ( u _ { w } ) is the limit at which exsorption will occur (so that \dot { s } < 0 ) . The transition between absorption and exsorption and vice versa takes place along “scanning” curves (discussed below). These curves are approximated by the single straight line shown in Figure 26.6.4–1.

When partial saturation is included in the analysis of flow through a porous medium, the absorption behavior, the exsorption behavior, and the scanning behavior (between absorption and exsorption) should each be defined. Each of these behaviors is discussed below. If sorption is not defined at all, Abaqus/Standard assumes fully saturated flow ( s = 1 . 0 ) for all values of u _ { w } .

Strongly unsymmetric partially saturated flow coupled equations result from the definition of sorption. Therefore, Abaqus/Standard automatically uses its unsymmetric matrix storage and solution scheme (see “Defining an analysis,” Section 6.1.2) if you request partially saturated analysis (i.e., if sorption is defined).

line

saturation	pore pressure -Uw
0.0	-Uw
0.5	-Uw
1.0	-Uw

Figure 26.6.4–1 Typical absorption and exsorption behaviors.

Defining absorption and exsorption

Absorption and exsorption behaviors are defined by specifying the pore liquid pressure, u _ { w } (negative “capillary tension”), as a function of saturation. In most physical cases the wetting liquid cannot be driven to zero saturation; to achieve zero saturation, the data would have to define u _ { w } \to - \infty { \mathrm { a s } } s \to 0 . 0 . Absorption and exsorption data can be defined in either a tabular form or an analytical form.

Tabular form

By default, you define the absorption and exsorption behaviors by specifying u _ { w } as a tabular function of \pmb { \mathscr { s } } , where 0 ^ { + } \leq s \leq 1 . 0 .

Input File Usage: Use the following options:

* { \mathrm { S O R P T I O N } } , { \mathrm { T Y P E } } { = } \mathrm { A B S O R P T I O N } , { \mathrm { L A W } } { = } { \mathrm { T A B U L A R } }
\ast \mathrm { S O R P T I O N } , \mathrm { T Y P E = E X S O R P T I O N } , \mathrm { L A W = T A B U L A R }

If the *SORPTION option is used only once, the behavior defined is taken as the behavior for absorption and exsorption.

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Sorption

Absorption: Law: Tabular

Exsorption: toggle on Include exsorption: Law: Tabular

Analytical form

The absorption and exsorption behaviors can be defined by the following analytical form:

\begin{array}{l} u _ {w} = \frac {1}{B} \mathrm{ln} \left[ \frac {(s - s _ {0})}{(1 - s _ {0}) + A (1 - s)} \right] \quad \text { for } \quad s _ {1} \leq s <   1, \\ u _ {w} = \left. u _ {w} \right| _ {s _ {1}} - \left. \frac {d u _ {w}}{d s} \right| _ {s _ {1}} (s _ {1} - s) \qquad \text {for} \quad s _ {0} \leq s <   s _ {1}, \\ \end{array}

where are positive material constants and s _ { 0 } , s _ { 1 } are parameters used to define the lower bound of the saturation values of interest (see Figure 26.6.4–2).

Figure 26.6.4–2 Logarithmic form of absorption and exsorption behaviors.

Input File Usage:	Use the following options:*SORPTION, TYPE=ABSORPTION, LAW=LOG*SORPTION, TYPE=EXSORPTION, LAW=LOGIf the *SORPTION option is used only once, the behavior defined is taken as the behavior for absorption and exsorption.
Abaqus/CAE Usage:	Property module: material editor: Other→Pore Fluid→SorptionAbsorption: Law: LogExsorption: toggle on Include exsorption: Law: Log

Defining the behavior between absorption and exsorption

The behavior between absorption and exsorption is defined by a scanning line of user-specified constant slope, \left( d u _ { w } / d s \right) \rvert _ { s } . This slope should be larger than the slope of any segment of the absorption or exsorption behaviors.

If absorption and exsorption behaviors are defined with no scanning line, the slope of the scanning line is taken as 1.05 times the largest value of d u _ { w } / d s given in the absorption and exsorption behavior definitions.

Input File Usage:

*SORPTION, TYPE=SCANNINGThis must be a repeated use of the *SORPTION option for the same material.

Abaqus/CAE Usage:

Property module: material editor: Other→Pore Fluid→Sorption: Exsorption: toggle on Include exsorption and Include scanning: Slope $(du_{w}/ds)|_{s}$

Elements

Sorption can be used only in elements that allow for pore pressure (see “Choosing the appropriate element for an analysis type,” Section 27.1.3).

Products: Abaqus/Standard Abaqus/CAE

References

• “Pore fluid flow properties,” Section 26.6.1
• “Material library: overview,” Section 21.1.1
• *GEL
• “Defining a swelling gel” in “Defining a fluid-filled porous material,” Section 12.12.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide

Overview

The swelling gel model:

• allows for modeling of the growth of gel particles that swell and trap wetting liquid in a partially saturated porous medium;
• is intended for use in moisture absorption problems, which typically involve polymeric materials, such as in the analysis of diapers; and
• can be used in the analysis of coupled pore liquid flow and porous medium stress (see “Coupled pore fluid diffusion and stress analysis,” Section 6.8.1).

Swelling gel model

The simple swelling gel model is based on the idealization of a gel as a volume of individual spherical particles of equal radius, r _ { a } . The swelling evolution (discussed in detail in “Constitutive behavior in a porous medium,” Section 2.8.3 of the Abaqus Theory Guide) is assumed to be given by

\dot {r} _ {a} = \frac {r _ {a} ^ {f} - r _ {a}}{\tau_ {1}} \left\langle s - 1 + \left(\frac {(r _ {a} ^ {f}) ^ {3} - (r _ {a}) ^ {3}}{(r _ {a} ^ {f}) ^ {3} - (r _ {a} ^ {\mathrm{dry}}) ^ {3}}\right) \right\rangle \left(1 - \left\langle \frac {r _ {a} - r _ {a} ^ {t}}{r _ {a} ^ {s} - r _ {a} ^ {t}} \right\rangle^ {2}\right),

where the value of any grouping of terms in angled brackets \langle \ \rangle is set equal to zero if its mathematical result is not positive, and

r_{a}^{f} is the fully swollen radius; \tau_{1} is the relaxation time of the gel particles; \pmb{s} is the saturation of the surrounding medium; r_{a}^{\mathrm{dry}} is the radius of the gel particles when they are completely dry; r_{a}^{t} = \left(\frac{n^{0}J}{4\sqrt{2}k_{a}}\right)^{\frac{1}{3}} is the maximum radius that the gel particles can achieve before they must touch; r_{a}^{s} = \left(\frac{3}{4\pi}\frac{n^{0}J}{k_{a}}\right)^{\frac{1}{3}} is the effective gel radius when the volume is entirely occupied with gel;

$n^{0}$	is the initial porosity of the material;
$J$	is the volume change in the material; and
$k_{a}$	is the number of gel particles per unit volume.

The second term in the definition of gel growth incorporates the assumption that the gel will swell only when the saturation of the surrounding medium, s, exceeds the effective saturation of the gel. The third term in the growth equation reduces the swelling rate when the surface of gel particles exposed to free fluid is limited by the combination of packing density and gel particle radius.

The swelling gel model is defined by specifying the variables r _ { a } ^ { \mathrm { d r y } } , r _ { a } ^ { f } , k _ { a } , and \tau _ { 1 }

Input File Usage: *GEL

Abaqus/CAE Usage: Property module: material editor: Other→Pore Fluid→Gel

Elements

The swelling gel model can be used only in elements that allow for pore pressure (see “Choosing the appropriate element for an analysis type,” Section 27.1.3).