Defining the Constitutive Response of Fluid Flow within the Cracked Element Surfaces

The tangential flow within the cracked element surfaces can be modeled with either a Newtonian or power-law model. By default, there is no tangential flow of pore fluid within the cracked element surfaces. To allow tangential flow, define a gap flow property in conjunction with the pore fluid material definition.

In the case of a Newtonian fluid the volume flow rate density vector is given by the expression

where $k_t$ is the tangential permeability (the resistance to the fluid flow), $\nabla p$ is the pressure gradient along the cracked element surfaces, and $d$ is the opening of the cracked element surfaces.

Abaqus defines the tangential permeability, or the resistance to flow, according to Reynold's equation:

where $\mu$ is the fluid viscosity and $d$ is the opening of the cracked element surfaces. You can also specify an upper limit on the value of $k_t$ .

In the case of a power law fluid the constitutive relation is defined as

where $\tau$ is the shear stress, $\dot{\gamma }$ is the shear strain rate, $K$ is the fluid consistency, and $\alpha$ is the power law coefficient. Abaqus defines the tangential volume flow rate density as

where $d$ is the opening of the cracked element surfaces.

By default, the gap between the cracked element surfaces has an initial opening of 0.002 in both a Newtonian fluid and a power law fluid. However, you can specify this opening directly.

GAP FLOW, TYPE=NEWTONIAN, KMAX

GAP FLOW, TYPE=POWER LAW

SECTION CONTROLS, INITIAL GAP OPENING

Property module: material editor: OtherPore FluidGap Flow: Type: Newtonian: Toggle on Maximum Permeability and enter the value of $k_{\max }$

Property module: material editor: OtherPore FluidGap Flow: Type: Power law

An initial gap opening is not supported in Abaqus/CAE.

You can permit normal flow by defining a fluid leak-off coefficient for the pore fluid material. This coefficient defines a pressure-flow relationship between the phantom nodes located at the cracked element edges and cracked element surfaces. The fluid leak-off coefficients can be interpreted as the permeability of a finite layer of material on the cracked element surfaces, as shown in Figure 1.

Figure 1. Leak-off coefficient interpretation as a permeable layer.

The normal flow is defined as

and

where $q_t$ and $q_b$ are the flow rates into the top and bottom surfaces of a cracked element, respectively; $p_i$ is the pressure at the phantom node located at the cracked element edge; and $p_t$ and $p_b$ are the pore pressures on the top and bottom surfaces of a cracked element, respectively. You can optionally define leak-off coefficients as functions of temperature and field variables.

Alternatively, you can use user subroutine UFLUIDLEAKOFF to define more complex leak-off behavior, including the ability to define a time accumulated resistance, or fouling, through the use of solution-dependent state variables.

FLUID LEAKOFF

FLUID LEAKOFF, DEPENDENCIES

FLUID LEAKOFF, USER

Property module: material editor: OtherPore FluidFluid Leakoff: Type: Coefficients

Property module: material editor: OtherPore FluidFluid Leakoff: Type: Coefficients: Toggle on Use temperature-dependent data and select the number of field variables.

Property module: material editor: OtherPore FluidFluid Leakoff: Type: User

If the thermal effect of the fluid flow within the cracked element surfaces is considered, the heat transport response comprises the tangential heat flux convection within the cracked surfaces, the normal heat convection between the gap fluid and the fracture surfaces, and the normal heat flux convected by fluid flow across the cracked element surfaces.

Optionally, you can include the normal heat convection between the gap fluid and the fracture surfaces.

The normal heat convection is defined as

and

where $q_t$ and $q_b$ are the heat fluxes into the top and bottom surfaces of a cracked element, respectively; ${\theta }_i$ is the temperature at the phantom node located on the cracked element edge; and ${\theta }_t$ and ${\theta }_b$ are the temperatures on the top and bottom surfaces of a cracked element, respectively.

SPECIFIC HEAT, PORE FLUID

CONDUCTIVITY, PORE FLUID

GAP CONVECTION

Abaqus allows the use of multiple fluids in an analysis, up to a maximum of four fluid types. This capability is useful in situations where the spatial distribution of the different fluid types is known (that is, precomputed) at all times during the analysis. In other words, the type of fluid at each node in the domain is known at all times during the analysis.

The definition of multiple fluids in an analysis involves associating each fluid with a unique integer and a specific predefined field variable. This integer acts as a fluid identifier. You can utilize this fluid identifier to specify the viscosity of each fluid. Only the fluid viscosity can be different for different fluids. All other properties (e.g., density, thermal) are assumed to be same for all the fluids. The field variable associated with a fluid type allows you to predefine the fluid type at a node as a function of time using amplitude definitions, or as functions of both time and space utilizing user-subroutines UFIELD or USDFLD.

You specify a value for each field variable, which is associated with a fluid identifier, at each node. It is recommended that you specify a value for the field variable ( $FV$ ) such that $0\le FV\le 1.0$ Abaqus assigns the fluid type at an integration point to be the one for which the field variable has the maximum value at that point. If all field variables have the same numerical value at an integration point, the first fluid (FLUID ID = 1) is assumed to be active.

*GAP FLOW, FLUID ID=Fluid Id number, DEFINITION=NEWTONIAN or POWER LAW or HERSCHEL-BULKLEY or BINGHAM PLASTIC
FIELD, VARIABLE=Field variable number
FLUID ID FIELD
Fluid Id number, Field variable number