### 26.5.7 Defining the constitutive response of fluid within the cohesive element gap

Products: ABAQUS/Standard  ABAQUS/CAE

### Overview

The cohesive element fluid flow model:

• is typically used in geotechnical applications, where fluid flow continuity within the gap and through the interface must be maintained;

• enables fluid pressure on the cohesive element surface to contribute to its mechanical behavior, which enables the modeling of hydraulically driven fracture; and

• enables modeling of an additional resistance layer on the surface of the cohesive element.

The features described in this section are used to model fluid flow within and across surfaces of pore pressure cohesive elements.

### Defining pore fluid flow properties

The fluid constitutive response comprises:

• Tangential flow within the gap, which can be modeled with either a Newtonian or power law model; and

• Normal flow across the gap, which can reflect resistance due to caking or fouling effects.

The flow patterns of the pore fluid in the element are shown in Figure 26.5.7–1. The fluid is assumed to be incompressible, and the formulation is based on a statement of flow continuity that considers tangential and normal flow and the rate of opening of the cohesive element.

### Specifying the fluid flow properties

You can assign tangential and normal flow properties separately.

#### Tangential flow

By default, there is no tangential flow of pore fluid within the cohesive element. To allow tangential flow, define a gap flow property in conjunction with the pore fluid material definition.

##### Newtonian fluid

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

where is the tangential permeability (the resistance to the fluid flow) and is the pressure gradient along the cohesive element.

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

where is the fluid viscosity and is the cohesive element opening. You can also specify an upper limit on the value of :

 Input File Usage: *GAP FLOW, TYPE=NEWTONIAN, KMAX

 ABAQUS/CAE Usage: Property module: material editor: OtherPore FluidGap Flow: Type: Newtonian: Toggle on Maximum Permeability and enter the value of

##### Power law fluid

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

where is the shear stress, is the shear strain rate, is the fluid consistency, and is the power law coefficient. ABAQUS defines the tangential volume flow rate density as

where is the cohesive element opening.

 Input File Usage: *GAP FLOW, TYPE=POWER LAW

 ABAQUS/CAE Usage: Property module: material editor: OtherPore FluidGap Flow: Type: Power law

#### Normal flow across gap surfaces

You can permit normal flow by defining a fluid leakoff coefficient for the pore fluid material. This coefficient defines a pressure-flow relationship between the cohesive element's middle nodes and their adjacent surface nodes. The fluid leakoff coefficients can be interpreted as the permeability of a finite layer of material on the cohesive element surfaces, as shown in Figure 26.5.7–2.

The normal flow is defined as

and

where and are the flow rates into the top and bottom surfaces, respectively; is the midface pressure; and and are the pore pressures on the top and bottom surfaces, respectively.

 Input File Usage: *FLUID LEAKOFF

 ABAQUS/CAE Usage: Property module: material editor: OtherPore FluidFluid Leakoff: Type: Coefficients

##### Defining leakoff coefficients as a function of temperature and field variables

You can optionally define leakoff coefficients as functions of temperature and field variables.

 Input File Usage: *FLUID LEAKOFF, DEPENDENCIES

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

##### Defining leakoff coefficients in a user subroutine

User subroutine UFLUIDLEAKOFF can also be used to define more complex leakoff behavior, including the ability to define a time accumulated resistance, or fouling, through the use of solution-dependent state variables.

 Input File Usage: *FLUID LEAKOFF, USER

 ABAQUS/CAE Usage: Property module: material editor: OtherPore FluidFluid Leakoff: Type: User

#### Tangential and normal flow combinations

You can define normal flow properties only if you are also defining tangential flow properties. Table 26.5.7–1 shows the permitted combinations and effects of each combination.

Table 26.5.7–1 Effects of flow property definition combinations.

Normal flow is definedNormal flow is undefined
Tangential flow is definedTangential and normal flow are modeled.Tangential flow is modeled. Pore pressure continuity is enforced between facing nodes in the cohesive element only when the element is closed. Otherwise, the surfaces are impermeable in the normal direction.
Tangential flow is undefined Normal flow is modeled.Tangential flow is not modeled. Pore pressure continuity is always enforced between facing nodes in the cohesive element.

### Initially open elements

When the opening of the cohesive element is driven primarily by entry of fluid into the gap, you will have to define one or more elements as initially open, since tangential flow is possible only in an open element. Identify initially open elements as initial conditions.

 Input File Usage: *INITIAL CONDITIONS, TYPE=INITIAL GAP

 ABAQUS/CAE Usage: Initial gap definition is not supported in ABAQUS/CAE.

### Use of unsymmetric matrix storage and solution

The pore pressure cohesive element matrices are unsymmetric; therefore, unsymmetric matrix storage and solution may be needed to improve convergence (see Matrix storage and solution scheme in ABAQUS/Standard” in “Procedures: overview, Section 6.1.1).

Your use of cohesive element fluid properties and your property values can impact your solution in some cases.

#### Large coefficient values

You must make sure that the tangential permeability or fluid leakoff coefficients are not excessively large. If either coefficient is many orders of magnitude higher than the permeability in the adjacent continuum elements, matrix conditioning problems may occur, leading to solver singularities and unreliable results.

#### Use in total pore pressure simulations

Definition of tangential flow properties may result in inaccurate results if the total pore pressure formulation is used and the hydrostatic pressure gradient contributes significantly to the tangential flow in the gap. The total pore pressure formulation is invoked if you apply gravity distributed loads to all elements in the model. The results will be accurate if the hydrostatic pressure gradient (i.e., the gravity vector) is perpendicular to the cohesive element.

### Output

The following output variables are available when flow is enabled in pore pressure cohesive elements:

 GFVR Gap fluid volume rate.
 PFOPEN Fracture opening.
 LEAKVRT Leak-off flow rate at element top.
 ALEAKVRT Accumulated leak-off flow rate at element top.
 LEAKVRB Leak-off flow rate at element bottom.
 ALEAKVRB Accumulated leak-off flow rate at element bottom.