Modeling Slurry Flow

Modeling discontinuities, such as cracks, as an enriched feature:

can include slurry transport and placement within the cracked element surfaces;
is typically used in geotechnical applications where the continuity of the slurry mixture and its components within the cracked element surfaces and across the cracked element surfaces must be maintained;
enables modeling of an additional resistance layer on the cracked element surfaces to model fluid leakoff into the formation, with or without considering the evolution of filter cake;
enables fluid pressure on the cracked element surfaces to contribute to its mechanical behavior, which enables the modeling of hydraulically driven fracture;
enables proppant settling due to gravity and wall drag effects;
enables the modeling of hydraulic fracture propped by solid particles; and
allows multiple fluid definition.

This page discusses:

Tangential Slurry Flow
Multifluid Slurry Flow
Proppant Settling
Crack Opening Particle Size Ratio
Stability of Tangential Flow
Normal Flow with Slurry Transport
Simple Leak-Off Behavior
Physics-Based Leak-Off Behavior
User Subroutine-Based Leak-off Behavior
Offset of Cracked Element Surfaces
Thermal Effect with Slurry Transport
Limits on Slurry Concentration
Simulating Flow of Clear Fluid
Output
References

Tangential Slurry Flow

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

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

where

d

is the opening of the cracked element surfaces,

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 (henceforth 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},

{\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 carrier fluid continuity equation 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 opening of the cracked element surfaces. 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 within the cracked surfaces (Defining the Constitutive Response of Fluid Flow within the Cracked Element Surfaces), 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

Multifluid Slurry Flow

Abaqus allows the use of multiple carrier fluids in an analysis that models slurry transport and placement. As described earlier (Defining Multiple Fluid Flows within the Cracked Surfaces), only the fluid viscosity can be different for different carrier fluids. All other properties (e.g., density, thermal) are assumed to be same for all the carrier fluids.

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 may 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 option 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 settling is not supported in Abaqus/CAE.

Crack Opening Particle Size Ratio

By default, Abaqus assumes that the ratio of the opening between the cracked element surfaces to the particle size must be greater than 1.0 for the slurry particles to enter the crack. Slurry calculations inside the cracked surfaces 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 crack opening to the particle size.

Input File Usage

Use the following option to define the density of the carrier fluid, the density of the proppant particles, and the diameter of the proppant particles (it is assumed that all the 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.

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 phenomenon dominated by advection. Although diffusion effects are assumed to be negligible, you can define a small amount of diffusivity to provide numerical stabilization.

Input File Usage

Use the following options 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.

A small amount of diffusivity is generally recommended for a slurry transport analysis. 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 proppants. Large values of

γ

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

Δ l

Normal Flow with Slurry Transport

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

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

The physics-based approach has 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 cracked element 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 with Slurry Transport).

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

Input File Usage

Use the following option to define the fluid leak-off coefficients at the top and the bottom surfaces 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 directly, (optionally) as tabular functions of 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 upon Outmans (1963) filtration models, which assume the following:

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 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 pore 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 mono-dispersed 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; you specify 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 may 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, it is computed internally by Abaqus 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 $Ω$ 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

Definition of physics-based leak-off model or corresponding user-subroutine are 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 crack surfaces and their derivatives with respect to the degrees of freedom 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 leak-off model is not supported in Abaqus/CAE.

Offset of Cracked Element Surfaces

A crack opening offset for contact of cracked element surfaces (see Defining Contact of Cracked Element Surfaces Using a Small-Sliding Formulation) 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 cracked element 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.

In an enriched element, an offset for contact between the cracked surfaces is specified using user subroutine UXFEMCRACK.

Thermal Effect with Slurry Transport

To model heat transfer associated with slurry transport within a crack, 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 computed automatically 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, Abaqus assumes the thermal conductivity is either a constant or a function of the temperature and field variables only.

CONDUCTIVITY, PORE FLUID, SLURRY

Abaqus/CAE Usage

Defining the heat capability and conductivity properties of the slurry fluid  is not supported in Abaqus/CAE.

Limits on Slurry Concentration

Depending upon 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 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 them. 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 non-default 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 the 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

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

Output

The following surface output variables are available only when the slurry fluid flow is enabled within the cracked enriched element surfaces:

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.
SLURRYVF: Volumetric concentration of proppant particles in slurry within the cracked surfaces in the enriched element.
SLURRYAF: Volume of proppant particles in slurry per unit area within the cracked surfaces in the enriched element.
THKFTCK: Filter cake thickness within the cracked surfaces in the enriched element.

References

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