ProppantTransport.rst

Proppant Transport Solver

Introduction

The ProppantTransport solver applies the finite volume method to solve the equations of proppant transport in hydraulic fractures. The behavior of proppant transport is described by a continuum formulation. Here we briefly outline the usage, governing equations and numerical implementation of the proppant transport model in GEOSX.

Theory

The following mass balance and constitutive equations are solved inside fractures,

Proppant-fluid Slurry Flow

$$\frac{\partial}{\partial t}(\rho_m) + \boldsymbol{\nabla} \cdot (\rho_m \boldsymbol{u_m}) = 0,$$

where the proppant-fluid mixture velocity u_m is approximated by the Darcy's law as,

$$\boldsymbol{u}_m = -\frac{K_f}{\mu_m}(\nabla p - \rho_m \boldsymbol{g}),$$

and p is pressure, ρ_m and μ_m are density and viscosity of the mixed fluid , respectively, and g is the gravity vector. The fracture permeability K_f is determined based on fracture aperture a as

$$K_f = \frac{a^2}{12}$$

Proppant Transport

$$\frac{\partial}{\partial t}(c) + \boldsymbol{\nabla} \cdot (c \boldsymbol{u}_p) = 0,$$

in which c and u_p represent the volume fraction and velocity of the proppant particles.

Multi-component Fluid Transport

$$\frac{\partial}{\partial t} [ \rho_i \omega_i (1 - c) ] + \boldsymbol{\nabla} \cdot [ \rho_i \omega_i (1 - c) \boldsymbol{u}_f ] = 0.$$

Here u_f represents the carrying fluid velocity. ρ_i and ω_i denote the density and concentration of i-th component in fluid, respectively. The fluid density ρ_f can now be readily written as

$$\rho_f = \sum_{i=1}^{N_c} \rho_i \omega_i,$$

where N_c is the number of components in fluid. Similarly, the fluid viscosity μ_f can be calculated by the mass fraction weighted average of the component viscosities.

The density and velocity of the slurry fluid are further expressed as,

ρ_m = (1 − c)ρ_f + cρ_p,

and

ρ_mu_m = (1 − c)ρ_fu_f + cρ_pu_p,

in which ρ_f and u_f are the density and velocity of the carrying fluid, and ρ_p is the density of the proppant particles.

Proppant Slip Velocity

The proppant particle and carrying fluid velocities are related by the slip velocity u_slip,

u_slip = u_p − u_f.

The slip velocity between the proppant and carrying fluid includes gravitational and collisional components, which take account of particle settling and collision effects, respectively.

The gravitational component of the slip velocity u_slipG is written as a form as

u_slipG = F(c)u_settling,

where u_settling is the settling velocity for a single particle, d_p is the particle diameter, and F(c) is the correction factor to the particle settling velocity in order to account for hindered settling effects as a result of particle-particle interactions,

F(c) = e^− λ_sc,

with the hindered settling coefficient λ_s as an empirical constant set to 5.9 by default (Barree & Conway, 1995).

The settling velocity for a single particle, u_settling , is calculated based on the Stokes drag law by default,

$$\boldsymbol{u}_{settling} = ( \rho_p - \rho_f) \frac{d{_p}^{2}}{18 \mu_f}\boldsymbol{g}.$$

Single-particle settling under intermediate Reynolds-number and turbulent flow conditions can also be described respectively by the Allen's equation (Barree & Conway, 1995),

$$\boldsymbol{u}_{settling} = 0.2 d_{p}^{1.18} \left [ \frac{g ( \rho_p - \rho_f)}{\rho_f} \right ]^{0.72} \left ( \frac{\rho_f}{\mu_f} \right )^{0.45} \boldsymbol{e},$$

and Newton's equation(Barree & Conway, 1995),

$$\boldsymbol{u}_{settling} = 1.74 d{_p}^{0.5}\left [ \frac{g ( \rho_p - \rho_f)}{\rho_f}\right]^{0.5} \boldsymbol{e}.$$

e is the unit gravity vector and d_p is the particle diameter.

The collisional component of the slip velocity is modeled by defining λ, the ratio of the particle velocity to the volume averaged mixture velocity as a function of the proppant concentration. From this the particle slip velocity in horizontal direction is related to the mixed fluid velocity by,

$$\boldsymbol{u}_{slipH} = \frac{\lambda - 1}{1 - c} \boldsymbol{v}_{m}$$

with v_m denoting volume averaged mixture velocity. We use a simple expression of λ proposed by Barree & Conway (1995) to correct the particle slip velocity in horizontal direction,

λ = [α−|c−c_slip|^β] 

where α and β are empirical constants, c_slip is the volume fraction exhibiting the greatest particle slip. By default the model parameters are set to the values given in (Barree & Conway, 1995): α = 1.27, c_slip = 0.1 and β = 1.5. This model can be extended to account for the transition to the particle pack as the proppant concentration approaches the jamming transition.

Proppant Bed Build-up and Load Transport

In addition to suspended particle flow the GEOSX has the option to model proppant settling into an immobile bed at the bottom of the fracture. As the proppant cannot settle further down the proppant bed starts to form and develop at the element that is either at the bottom of the fracture or has an underlying element already filled with particles. Such an "inter-facial" element is divided into proppant flow and immobile bed regions based on the proppant-pack height.

Although proppant becomes immobile fluid can continue to flow through the settled proppant pack. The pack permeability K is defined based on the Kozeny-Carmen relationship:

$$K = \frac{(sd_p)^2}{180}\frac{\phi^{3}}{(1-\phi)^{2}}$$

and

ϕ = 1 − c_s

where ϕ is the porosity of particle pack and c_s is the saturation or maximum fraction for proppant packing, s is the sphericity and d_p is the particle diameter.

The growth of the settled pack in an "inter-facial" element is controlled by the interplay between proppant gravitational settling and shear-force induced lifting as (Hu et al., 2018),

$$\frac{d H}{d t} = \frac{c u_{settling} F(c)}{c_{s}} - \frac{Q_{lift}}{A c_{s}},$$

where H, t, c_s, Q_lift, and A represent the height of the proppant bed, time, saturation or maximum proppant concnetration in the proppant bed, proppant-bed load (wash-out) flux, and cross-sectional area, respectively.

The rate of proppant bed load transport (or wash out) due to shear force is calculated by the correlation proposed by Wiberg and Smith (1989) and McClure (2018),

$$Q_{lift} = a \left ( d{_p} \sqrt{\frac{g d{_p} ( \rho_p - \rho_f)}{\rho_f}} \right ) (9.64 N_{sh}^{0.166})(N_{sh} - N_{sh, c})^{1.5}.$$

a is fracture aperture, and N_sh is the Shields number measuring the relative importance of the shear force to the gravitational force on a particle of sediment (Miller et al., 1977; Biot & Medlin, 1985; McClure, 2018) as

$$N_{sh} = \frac{\tau}{d{_p} g ( \rho_p - \rho_f)},$$

and

τ = 0.125fρ_fu_m²

where τ is the shear stress acting on the top of the proppant bed and f is the Darcy friction coefficient. N_sh, c is the critical Shields number for the onset of bed load transport.

Proppant Bridging and Screenout

Proppant bridging occurs when proppant particle size is close to or larger than fracture aperture. The aperture at which bridging occurs, h_b, is defined simply by

h_b = λ_bd_p,

in which λ_b is the bridging factor.

Slurry Fluid Viscosity

The viscosity of the bulk fluid, μ_m, is calculated as a function of proppant concentration as (Keck et al., 1992),

$$\mu_{m} = \mu_{f}\left [1 + 1.25 \left ( \frac{c}{1-c/c_{s}} \right) \right ]^{2}.$$

Note that continued model development and improvement are underway and additional empirical correlations or functions will be added to support the above calculations.

Spatial Discretization

The above governing equations are discretized using a cell-centered two-point flux approximation (TPFA) finite volume method. We use an upwind scheme to approximate proppant and component transport across cell interfaces.

Solution Strategy

The discretized non-linear slurry flow and proppant/component transport equations at each time step are separately solved by the Newton-Raphson method. The coupling between them is achieved by a time-marching sequential (operator-splitting) solution approach.

Parameters

The solver is enabled by adding a <ProppantTransport> node and a <SurfaceGenerator> node in the Solvers section. Like any solver, time stepping is driven by events, see EventManager.

The following attributes are supported:

In particular:

• discretization must point to a Finite Volume flux approximation scheme defined in the Numerical Methods section of the input file (see FiniteVolumeDiscretization)
• proppantName must point to a particle fluid model defined in the Constitutive section of the input file (see Constitutive)
• fluidName must point to a slurry fluid model defined in the Constitutive section of the input file (see Constitutive)
• solidName must point to a solid mechanics model defined in the Constitutive section of the input file (see Constitutive)
• targetRegions attribute is currently not supported, the solver is always applied to all regions.

Primary solution field labels are proppantConcentration and pressure. Initial conditions must be prescribed on these field in every region, and boundary conditions must be prescribed on these fields on cell or face sets of interest. For static (non-propagating) fracture problems, the fields ruptureState and elementAperture should be provided in the initial conditions.

In addition, the solver declares a scalar field named referencePorosity and a vector field named permeability, that contains principal values of the symmetric rank-2 permeability tensor (tensor axis are assumed aligned with the global coordinate system). These fields must be populated via FieldSpecification section and permeability should be supplied as the value of coefficientName attribute of the flux approximation scheme used.

Example

First, we specify the proppant transport solver itself and apply it to the fracture region:

/coreComponents/physicsSolvers/multiphysics/integratedTests/FlowProppantTransport_2d.xml

Then, we specify a compatible flow solver (currently a specialized SinglePhaseProppantFVM solver must be used):

/coreComponents/physicsSolvers/multiphysics/integratedTests/FlowProppantTransport_2d.xml

Finally, we couple them through a coupled solver that references the two above:

/coreComponents/physicsSolvers/multiphysics/integratedTests/FlowProppantTransport_2d.xml

References

• 18. 4. Barree & M. W. Conway. "Experimental and numerical modeling of convective proppant transport", JPT. Journal of petroleum technology, 47(3):216-222, 1995.
• 13. 1. Biot & W. L. Medlin. "Theory of Sand Transport in Thin Fluids", Paper presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, NV, 1985.
• 24. Hu, K. Wu, X. Song, W. Yu, J. Tang, G. Li, & Z. Shen. "A new model for simulating particle transport in a low-viscosity fluid for fluid-driven fracturing", AIChE J. 64 (9), 35423552, 2018.
• 18. 7. Keck, W. L. Nehmer, & G. S. Strumolo. "A new method for predicting friction pressures and rheology of proppant-laden fracturing fluids", SPE Prod. Eng., 7(1):21-28, 1992.
• 13. McClure. "Bed load proppant transport during slickwater hydraulic fracturing: insights from comparisons between published laboratory data and correlations for sediment and pipeline slurry transport", J. Pet. Sci. Eng. 161 (2), 599610, 2018.
• 13. 3. Miller, I. N. McCave, & P. D. Komar. "Threshold of sediment motion under unidirectional currents", Sedimentology 24 (4), 507527, 1977.
• 16. 12. Wiberg & J. D. Smith. "Model for calculating bed load transport of sediment", J. Hydraul. Eng. 115 (1), 101123, 1989.