Defining Slurry Transport and Placement within the Cohesive Element Gap

Modeling Slurry Flow

The slurry transport and placement capabilities in Abaqus/Standard model transport of solids (proppants)-laden Newtonian and non-Newtonian slurries within the gap between the two faces of a cohesive element. The slurry advects into the gap region, as it opens up due to hydraulic pressure. Subsequently, the particles in the slurry settle down within the gap through a process that is mediated by wall drag and gravity effects. The particles help prop open the gap if it tries to close down due to changes in external conditions.

This complex set of phenomena within the cracked surfaces can be described in terms of the mean tangential flow of the slurry mixture, the advection of the proppant particles with the carrying fluid, and the settling of the proppant particles within the gap due to wall drag and gravity effects. The process is governed by the continuity of the slurry mixture (a mixture of the proppant particles and the carrying fluid) that describes the mean flow of the slurry, the continuity of the carrier fluid, a description of the settling phenomenon in terms of the relative velocity of the particles with respect to the carrying fluid, and leakoff (normal flow) of the carrying fluid into the surrounding formation.

Tangential Slurry Flow

The continuity equation for the mean flow of the slurry mixture within the gap can be written as

\frac{\partial d}{\partial t} + \nabla\cdot (d V_{m}) + q_{l} = 0,

where

d

is the gap opening,

V_{m}

is the mean velocity of slurry, and

q_{l}

is the leak-off flow rate. The continuity equation for the carrier fluid can be written as

\frac{\partial [(1 - C_{v}) d]}{\partial t} + \nabla\cdot [(1 - C_{v}) d V_{w}] + q_{l} = 0,

where

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 referred to as the slurry concentration), and

V_{w}

is the velocity vector of the carrier fluid. In formulating the above equation, it is assumed that the carrier fluid and solid particles move with different velocity vectors,

V_{w}

and

V_{s}

, respectively, and that they are related to the mean slurry velocity and a settling (or slip) velocity,

V_{s / w},

as

{\begin{matrix} V_{w} = V_{m} + C_{v} V_{s / w} \\ V_{s} = V_{m} - (1 - C_{v}) V_{s / w} \end{matrix},

where the settling velocity

V_{s / w} = V_{w} - V_{s}

represents the relative velocity of the proppant particles with respect to the carrier fluid. Using the above velocity split, the continuity equation for the carrier fluid can be rewritten as

\frac{\partial [(1 - C_{v}) d]}{\partial t} + \nabla\cdot [(1 - C_{v}) d V_{m}] + q_{l} + \nabla\cdot [C_{v} (1 - C_{v}) d V_{s / w}] = 0.

The above system of equations is augmented by a relation of the following form, which relates the mean slurry flux, $q$ , and the gradient of the fluid pressure field (assuming non-Newtonian constitutive behavior of the slurry):

q d = f (K, α, τ_{0}, d, \nabla p),

where

K

is the fluid consistency,

α

is the power law coefficient,

τ_{0}

is the yield stress, and

d

is the gap opening. The above system of equations helps determine the fluid pressure field that drives the slurry flow through the gap, the mean slurry flow velocity field, and the slurry concentration,

C_{v}

(nodal degree of freedom 31), accounting for both leakoff and settling.

Abaqus assumes that the homogenized slurry flow properties can be defined in a manner that is similar to the corresponding definitions for the flow of a pure fluid with the cohesive gap (Defining the Constitutive Response of Fluid Transitioning from Darcy Flow to Poiseuille Flow), except that the flow properties can be functions of the slurry concentration.

Input File Usage

Use the following option to define the tangential flow of a slurry mixture with a specified fluid type:

GAP FLOW, TYPE=type_name, SLURRY

Abaqus/CAE Usage

Defining the tangential flow of the slurry mixture is not supported in Abaqus/CAE.

Proppant Settling

By default, Abaqus assumes that the particle settling phenomena is mainly due to gravity (solid particles and carrier fluid usually have different densities) and wall drag effects (hindrance due to the narrowness of the gap). However you can suppress the settling completely (thereby modeling proppant transport due to advection only), or you can define the settling velocity field directly through user subroutine USETTLING.

Input File Usage

Use one of the following options to account for settling effects using the Abaqus built-in model:

GAP FLOW, TYPE=type_name, SLURRY

GAP FLOW, TYPE=type_name, SLURRY, SETTLING

Use the following option to suppress settling effects:

GAP FLOW, TYPE=type_name, SLURRY, SETTLING=NONE

Use the following option to define the settling velocity field directly through user subroutine USETTLING:

GAP FLOW, TYPE=type_name, SLURRY, SETTLING=USER

Abaqus/CAE Usage

Specifying proppant particle settling is not supported in Abaqus/CAE.

Gap Particle Size Ratio

By default, Abaqus assumes that the ratio of the opening between the gap surfaces to the particle size must be greater than 1.0 for the slurry particles to enter the gap. Slurry calculations inside the gap are not activated by default if the calculated ratio between these quantities is less than 1.0. Alternatively, you can specify a nondefault minimum ratio of the gap opening to the particle size.

Input File Usage

Use the following option to define the density of carrier fluid, the density of the proppant particles, and the diameter of proppant particles (it is assumed that all proppant particles in the slurry have the same diameter), respectively:

DENSITY, SLURRY

Use the following option to define the ratio of the minimum gap opening to the particle size directly:

SECTION CONTROLS, MIN GAP PARTICLE RATIO

Abaqus/CAE Usage

Defining the density of the carrier fluid and the proppant particles and the gap-particle diameter ratio of the slurry is not supported in Abaqus/CAE.

Multiple Fluid Flow

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 for each fluid type,and associating each fluid identifier with a specific predefined field variable used to specify the spatial distribution of the fluid as a 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 (for example, density, thermal, etc.) 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 ( $F V$ ) such that $0 \leq F V \leq 1.0$ . 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 1) is assumed to be active.

Input File Usage

*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

Abaqus/CAE Usage

Specifying multiple fluid flow is not supported in Abaqus/CAE.

Stability of Tangential Flow

A Petrov-Galerkin scheme (see Convection/diffusion) is used to discretize the governing equation for the slurry concentration because the slurry transport is a highly advection dominant phenomenon. Although diffusion effects are assumed to be negligible, you can define a small amount of diffusivity to provide numerical stabilization.

You must be careful to select a value of the diffusivity coefficient such that the fundamental nature of the transport problem is not modified from convection dominated to diffusion dominated. A good measure of the relative dominance of convective versus diffusive contributions is the local Péclet number, $γ$ , which is defined as:

γ = | v | \frac{Δ l}{D},

where

| v |

is the magnitude of the velocity vector,

Δ l

is a characteristic element length in the direction of flow, and

D

is the diffusivity of the proppant. Large values of

γ

indicate that convection dominates over diffusion on the spatial scale defined by the element size

Δ l

.

Input File Usage

Use the following option to define the diffusivity of proppant particles in a slurry flow:

DIFFUSIVITY, SLURRY

Abaqus/CAE Usage

Defining the diffusivity of proppant particles of a slurry flow is not supported in Abaqus/CAE.

Normal Flow across Gap Surfaces

The normal flow across the gap surfaces can be modeled using one of the following approaches:

A simple leak-off behavior that requires leak-off coefficients on the top and bottom surfaces to define the flux of the normal flow.
A physics-based filtration model that accounts for the formation and evolution of filter cake layers on the two cohesive surfaces.

The physics-based approach has the following flavors (Outmans, 1963), which are applicable only when the slurry concentration degree of freedom is active in the analysis:

A dynamic filtration model that assumes the filter cake to be compressible.
A static filtration model that assumes the filter cake to be incompressible.

Therefore, you can model the normal flow across the gap surfaces using one of the following approaches:

the simple leak-off model in which you define the leak-off coefficients directly;
a physics-based model that accounts for the evolution of the filter cake thickness, filter cake viscosity, and filter case permeability; or
- through user subroutine UFLUIDLEAKOFF, which allows you 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).

Simple Leak-Off Behavior

The simple leak-off behavior assumes that the flux of the normal flow across the top and the bottom surfaces is defined by a simple pressure-flow relationship (for more information, see Normal Flow across Gap Surfaces).

The simple leak-off model does not account for the dependence of the leak-off behavior on the slurry concentration but can still provide a reasonable description of the normal flow across gaps in some situations.

Input File Usage

Use the following option to define the fluid leak-off coefficients at the top and the bottom surfaces of the cohesive element directly, (optionally) as tabular functions of temperature:

FLUID LEAKOFF

Use the following option to define the fluid leak-off coefficients at the top and the bottom surfaces of the cohesive element directly, (optionally) as tabular functions of (optionally) temperature and field variables:

FLUID LEAKOFF, DEPENDENCIES

Use the following option to define the fluid leak off coefficients using user subroutine UFLUIDLEAKOFF:

FLUID LEAKOFF, USER

Abaqus/CAE Usage

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

Physics-Based Leak-Off Behavior

The physics-based leak-off behavior is based on Outmans (1963) filtration models, which assume:

the filter cake is a porous medium;
the fluid flow through the filter cake is governed by Darcy’s law; and
the rate of increase of the thickness of the filter cake layer is proportional to the flux of fluid through the filter cake.

Abaqus assumes that the filter cake to be compressible in a dynamic filtration model, while it is assumed to be incompressible in a static filtration model.

In the discussion to follow, $p_{f}$ and $p_{p}$ represent the (slurry) fluid pressure and formation pope fluid pressure, respectively.

For the dynamic filtration model, the porosity of the filter cake is defined as

ε_{h} = ε_{0} \exp [- α (p_{f} - p_{p})],

where

ε_{0} = C_{f} ε^{*},

ε^{*} = 0.68,

is the porosity corresponding to packing of monodispersed spherical particles,

C_{f}

represents a correction factor that accounts for particle size variations, and

α

is a compressibility factor. The case

α = 0

represents incompressible static filter cake behavior, which implies that

ε_{h} = ε_{0} .

You specify both

C_{f}

and

α

to define a dynamic filtration model and only

C_{f}

to define a static filtration model.

The porosity associated with the slurry is defined in terms of the slurry concentration as $ε_{s} = 1 - C_{v} .$ The rate of growth of the thickness of the filter cake layer is assumed to be

\frac{d h}{d t} = b \frac{κ}{μ} \frac{\partial p}{\partial x},

where

b = \frac{1 - ε_{s}}{ε_{s} - ε_{h}}

is the specific filter cake volume,

p

represents the fluid pressure in the filter cake, and

x

represents the coordinate in the thickness direction of the filter cake. You can optionally specify an initial value of the filter cake thickness,

h_{0} .

The above equation is integrated in time to obtain the thickness of the filter cake layer,

h (t),

as a function of time. In the incompressible case, the specific filter cake volume simplifies to

b = \frac{C_{v}}{1 - C_{v} - C_{f} ε^{*}} .

The permeability of the filter cake is computed as

κ = C_{d} \frac{ε_{h}}{1 - ε_{h}},

where

C_{d}

is a permeability factor that you can specify directly. If you do not specify the permeability factor, Abaqus computes it internally as

C_{d} = \frac{ε_{0}^{2} D_{p}^{2}}{150 (1 - ε_{0})},

where

D_{p}

is the particle diameter. In the case of an incompressible filter cake (that is,

α = 0

), the expression for permeability becomes

C_{d} = \frac{ε_{0}^{3} D_{p}^{2}}{150 {(1 - ε_{0})}^{2}},

which is very similar in form to the well-known Kozeny-Carman equation describing the permeability for a fluid flowing through a packed bed of solids.

The flux of the normal flow across the cohesive surfaces is

q_{t} = - \frac{κ}{μ} \frac{(p_{f} - p_{p})}{h (t)} Ω,

where

κ

and

μ

represent the permeability of the filter cake and the filtrate viscosity, respectively;

h (t)

is the time-dependent thickness of the filter cake layer; and

Ω (\leq 1.0)

is a numerical factor that accounts for the compressibility of the filter cake layer. For the dynamic filtration model, Abaqus computes the factor

Ω

internally based on

α

; while for the static filtration model,

Ω = 1.0

.

Input File Usage

Use the following option to define the physics-based leak-off model:

FLUID LEAKOFF, SLURRY

Abaqus/CAE Usage

Defining a physics-based leak-off model is not supported in Abaqus/CAE.

User Subroutine-Based Leak-off Behavior

You can use user subroutine UFLUIDLEAKOFF to define a physics-based model of your choice. In this case, you define the fluid flux across the top and the bottom surfaces of the cohesive elements and their derivatives with respect to the degrees of freedom of the cohesive element directly in user subroutine UFLUIDLEAKOFF.

Input File Usage

Use the following option to define more complex leak-off behavior in user subroutine UFLUIDLEAKOFF:

FLUID LEAKOFF, SLURRY, USER

Abaqus/CAE Usage

Defining a user-specified leakoff model is not supported in Abaqus/CAE.

Specifying Thermal Properties of Slurry Flow

To model heat transfer associated with the slurry transport within a gap, you must define the relevant thermal properties. Optionally, the thermal properties can be defined as functions of the slurry concentration. In particular, the thermal conductivity can be defined as a function of the slurry concentration directly in a tabular form. However, the specific heat of the slurry is automatically computed based on the specific heats of the carrying fluid and the proppant particles, respectively, using a rule of mixtures.

Input File Usage

Use the following option to specify the heat capacity as a function of the slurry concentration. If the SLURRY parameter is omitted, Abaqus assumes the heat capacity is either a constant or a function of the temperature and field variables only.

SPECIFIC HEAT, PORE FLUID, SLURRY

Use the following option to specify the thermal conductivity of the slurry as a function of the slurry concentration. If the SLURRY parameter is omitted, we assume that the thermal conductivity is either a constant or a function of the temperature and field variables only.

CONDUCTIVITY, PORE FLUID, SLURRY

Abaqus/CAE Usage

Definitions of thermal properties of the slurry are not supported in Abaqus/CAE.

Cohesive Offset

A cohesive offset (see Cohesive Offset) refers to a value of the separation in the normal direction, $δ_{n} = δ_{n}^{o f f}$ , below which the resistance to additional closure $(δ_{n} > 0, {\dot{δ}}_{n} < 0)$ of the cohesive surfaces increases significantly. This capability is useful in preventing the closure of a fracture when changes in external loading conditions might otherwise result in closure. A practical application of this capability occurs during hydraulic fracturing with slurry, where solid proppant particles (the solid part of the slurry) help "prop open" the fracture when external hydraulic pressure loads are removed.

Limits on Slurry Concentration

Depending on the nature of packing of the proppant particles (assumed to be spherical), there will be a theoretical upper limit on the slurry concentration. By default, Abaqus assumes this upper limit to be 0.67. The lower limit of the slurry concentration is assumed to be 0. The strongly advective nature of the slurry transport problem that can lead to oscillations in the solution; therefore, these theoretical limits might be violated numerically in some cases. The use of a Pertov Galerkin scheme to formulate the slurry transport continuity equation helps alleviate such numerical oscillations to a large extent but does not completely eliminate it. As discussed earlier (see Stability of Tangential Flow), the use of artificial diffusivity is recommended for most applications.

You can specify nondefault upper and lower limits on the slurry concentration.

Input File Usage

Use the following option to specify nondefault upper and the lower limits on the slurry concentration:

In a geostatic procedure:

GEOSTATIC, SLURRY MIN=min_value,SLURRY MAX=max_value

In a soils consolidation procedure:

SOILS, CONSOLIDATION, SLURRY MIN=min_value,SLURRY MAX=max_value

Abaqus/CAE Usage

Specifying nondefault upper and lower limits on the slurry concentration is not supported in Abaqus/CAE.

Simulating Flow of Clear Fluid

Some hydraulic fracturing applications involve the use of clear fluid to initiate fracture, followed by the use of a slurry. To model such applications, you can use multiple steps in the analysis and specify that Abaqus does not solve for slurry concentration during the step in which a clear fluid is used.

Input File Usage

Use the following option to define a flow of clear fluid:

In a geostatic procedure:

GEOSTATIC, SLURRY FLOW=NO

In a soils consolidation procedure:

SOILS, CONSOLIDATION, SLURRY FLOW=NO

Abaqus/CAE Usage

Defining a flow of clear fluid is not supported in Abaqus/CAE.

Output

In addition to the output variables that are available with fluid flow-enabled cohesive elements, the following variables are available when slurry transport is modeled:

SLURRYVF: Volumetric concentration of proppant particles in the slurry.
SLURRYAF: Volume of proppant particles in the slurry per unit area.
THKFTCKT: Filter cake thickness with leak-off at top surface.
THKFTCKB: Filter cake thickness with leak-off at bottom surface.
FLVF: When multiple fluids are defined, the components of FLVF represent the volume fraction of each fluid type. In particular, FLVF_i refers to the volume fraction of fluid type i.

References

Outmans, H. D., “Mechanics of Static and Dynamic Filtration In the Borehole,” Society of Petroleum Engineers Journal, vol. 3, pp. 236–244, 1963.