Slurry Fluid Pipe Elements

You can use slurry fluid pipe elements and thermal-slurry fluid pipe elements in models that use symmetry to reduce the model size. These elements model circular geometry. You can specify a symmetry value that is greater than 0 and less than or equal to 1. This value is interpreted as scaling of the circular pipe geometry in radians. A value of 1 corresponds to $2\pi$ radians, and a value of 0.5 corresponds to $\pi$ radians. When you specify symmetry, you must scale the full model flow magnitude by the symmetry value. You do not need to apply any symmetric constraints or boundary conditions.

FLUID PIPE SECTION, SYMMETRY=value

Fluid Pipe Equations describes the governing equations for the fluid flow and pressure drop computation for fluids in pipes. Specifying the Friction Loss Behavior describes the different choices of specifying the frictional loss characteristics for the flowing fluid in the thermal pipe. You can model additional fluid pressure losses in thermal fluid pipe elements (see Additional Loss Terms in Fluid Pipe Elements). The governing equations to model temperature evolutions are described in Thermal Fluid Pipe Equations. When using the thermal slurry fluid pipe elements, you can specify the pipe wall section and convection behavior as described in Specifying the Wall Diameters for Thermal Fluid Pipe Elements and Specifying the Type of Convection between the Fluid and the Pipe Wall.

When slurry flows through the tubing, pipe, or annulus, the flow conditions might change and lead to heterogeneous slurry flow and settling of solids. This can be observed in slurry transport through long horizontal sections at flow velocities too low to maintain the solids in suspension. Under horizontal flow conditions and the influence of gravity, significant gradation in the concentration of solids can develop in the form of various slurry flow regimes (for example, heterogeneous, saltation, or sliding bed). The presence of solid particles in the slurry generates additional momentum transfer between the particles and pipe wall, in addition to the transfer between the carrier fluid and pipe wall. The net effect of the additional transfer mechanism is an increase in pressure drop. Based on the concept of additive pressure drop, the fractional increase in pressure gradient due to the presence of solid particles can be added to that produced by interaction of the carrier fluid with the pipe wall:

• $\mathrm{\Delta }{P}_{f,m}$ is the pressure drop of the mixture,
• $\mathrm{\Delta }{P}_{f,w}$ is the pressure drop due to the flow of the carrier fluid alone, and
• $\mathrm{\Delta }{P}_{f,s}$ is the pressure drop due to the flow of solid particles.

Based on many experimental observations, an extended pressure loss definition was proposed by Turian and Yuan (1977), to account for various flow regimes often prevailing in the transport of slurry depending upon flow conditions. The pressure gradient in each flow regime can be correlated using an equation of the form:

• $C_v$ is the concentration of proppant particles in the slurry, defined as the volume ratio of the solid phase to that of the slurry phase (subsequently simply referred to as the slurry concentration);
• $f_w$ is the friction factor due to the carrier fluid flow;
• $C_D^{*}$ is the drag coefficient of a single sphere settling at terminal velocity in an unbounded quiescent fluid; and
• $Fr$ is the pipe Froude number modified to account for particle density ${\rho }_s$ and is defined as $\left[\frac{V^2}{g{D}_P(\frac{{\rho }_s}{{\rho }_w}-1)}\right]$ , where $V$ is the flow velocity, $g$ is gravity, and $D_P$ is the particle diameter.

Based on the work of Turian and Yuan (1977), the values of these coefficients were defined for each of the slurry flow regimes, namely sliding bed (Regime 0), saltation (Regime 1), heterogeneous suspension (Regime 2) and homogeneous suspension (Regime 3):

Sliding Bed (Regime 0 - $R_0$ ):

Saltation (Regime 1 - $R_1$ ):

Heterogeneous suspension (Regime 2 - $R_2$ ):

Homogeneous suspension (Regime 3 - $R_3$ ):

Demarcation of the slurry flow regimes is accomplished by ensuring that any two regimes are contiguous at their common boundary. This implies that any two regime correlation equations must be satisfied simultaneously. For example, the boundary demarcating the sliding bed and saltation regimes must lie along the locus of the equation for regimes $R_0=R_1$ . The transition number demarcating the boundary between the sliding bed ( $R_0$ ) and saltation ( $R_1$ ) regimes can be written as

Similarly, the equations for other regime transitions can be written as

An important aspect of implementing the approach developed by Turian et al., is a process by which the flow regime can be readily established. It is possible to identify the applicable regime for a given set of flowing conditions just from knowledge of the transition numbers $R_{ij}$ . For the transition numbers $R_{ij}$ , where $i<j$ , the transition number $R_{ij}$ increases monotonically as the flow velocity increases. At low flow velocities, $R_{ij}<1$ and eventually passes through the value 1.0 with increasing flow velocity. This indicates a transition out of Regime i. For any combination of average flow velocity, $V$ , and particle diameter $D_p$ , the flow regime is fixed according to the constraints: $R_{ij}<1$ , the regime is not j; and $R_{ij}>1$ , the regime is not i. For further details on the algorithm, refer to Turian and Yuan (1977).

After the drag coefficient, friction factor, and pipe Froude number are calculated, the transition numbers are calculated and tested to identify the correct slurry flow regime. Once the slurry flow regime is established, the slurry friction factor and fractional loss due to solids are determined.

You must associate a material definition with each pipe element section property.

The material that is defined for the thermal-slurry fluid pipe section refers to the slurry that is flowing through the pipe and the thermal properties for the pipe wall. You must define the fluid density and the viscosity of the slurry to describe its flow behavior. The density of the slurry can be specified to be a function of the slurry concentration. In this case, the density is computed using the density of the fluid and the solid proppant particles, and the slurry concentration, using a rule of mixtures. The viscosity of the slurry can be specified as a function of the slurry concentration directly in tabular form. For the viscosity definition, slurry fluid pipe elements support both Newtonian and non-Newtonian fluids. The following non-Newtonian fluids are supported: power law, Bingham Plastic, and Herschel-Bulkley models (see Viscosity).

In addition, for the thermal-slurry fluid pipe elements, you must specify the following thermal properties:

• Conductivity and specific heat of the slurry. The conductivity can optionally be specified as a function of the slurry concentration (as well as temperature and field variables, if necessary) directly in tabular form. The specific heat can also be specified to be dependent upon the slurry concentration. In this case the specific heat of the slurry is computed using a rule of mixtures based upon the specific heat of the carrying fluid, the specific heat of the solid proppant particles, and the slurry concentration.

FLUID PIPE SECTION, MATERIAL=material name
MATERIAL, NAME=material name
VISCOSITY, DEFINITION=NEWTONIAN or POWER LAW or HERSCHEL-BULKLEY or BINGHAM PLASTIC, SLURRY
DENSITY, PORE FLUID, SLURRY
CONDUCTIVITY, PORE FLUID, SLURRY
SPECIFIC HEAT, PORE FLUID, SLURRY

DENSITY
CONDUCTIVITY
SPECIFIC HEAT

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 (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 assigning a unique integer value that acts as an identifier of each fluid type, and associating each fluid identifier with a specific predefined field variable used to specify the spatial distribution of the fluid as function of time. You can utilize the fluid identifier to specify the viscosity of each fluid. Only the fluid viscosity can be different for different fluids. All other properties (that is, density and 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 subroutine UFIELD or USDFLD.

You specify a value for each field variable, associated with a fluid identifier, at each node. It is recommended that you specify a value for the field variable ( $FV$ ) such that $0.0\le FV\le 1.0$ . 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 identifier equal to 1) is assumed to be active.

VISCOSITY, 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

You can define an initial slurry concentration, an initial temperature, and a field distribution over the nodes of the slurry fluid pipe element and thermal-slurry fluid pipe element.

INITIAL CONDITIONS, TYPE=TEMPERATURE
INITIAL CONDITIONS, TYPE=FIELD
INITIAL CONDITIONS, TYPE=SLURRYVF

Slurry fluid pipe elements allow for the specification of pressure boundary conditions and volumetric flow rates at the nodes. At a particular node, you can specify either a pressure or flow rate but not both. You can also specify a gravity load on the slurry fluid pipe element and thermal-slurry fluid pipe element to determine the hydrostatic head at the nodes. For the thermal-slurry fluid pipe elements, you can specify the fluid temperature at the inlet or outlet and at the pipe wall. You can specify the slurry concentration at the inlet or outlet of the node that captures the fluid pressure loss.

BOUNDARY
node or node set, 8, 8, magnitude

CFLOW
node or node set, , magnitude

BOUNDARY
node or node set, 11, 11, magnitude

BOUNDARY
node or node set, 31, 31, magnitude

DLOAD
element or element set, , GRAV, gravity constant, comp1, comp2, comp3