S. A. Boronin

Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.

Stability of a disperse-mixture flow in a boundary layer

The hydrodynamic stability of a dilute disperse mixture flow in a quasi-equilibrium region of a boundary layer with a significantly nonuniform particle concentration profile is investigated. The mixture is described by a two-fluid model with an incompressible viscous carrier phase. In addition to the Stokes drag, the Saffman lifting force is taken into account in the interphase momentum exchange. On the basis of a numerical solution of the boundary-value problem for a modified Orr-Sommerfeld equation, neutral stability curves are analyzed and the dependence of the critical Reynolds number on the governing parameters is studied. It is shown that taking into account the particle concentration nonuniformity in the main flow and the Saffman lifting force significantly changes the stability limits of the two-phase laminar boundary layer flow. The effect of these factors on the boundary layer stability is considered for the first time.

Effects of particle migration on suspension flow in a hydraulic fracture

An asymptotic model of a hydraulic-fracture flow of a sedimenting concentrated suspension is formulated on the basis of the two-fluid approach with account of transverse particle migration. In the thin-layer approximation, a two-dimensional system of equations averaged across the fracture is constructed with account for a nonuniform distribution of the particle concentration. As compared to the similar model without particle migration, the averaged two-dimensional equations contain modified coefficients which explicitly depend on the width of the flow core occupied by the particles. Using the model constructed, a numerical simulation is performed, which shows that the particle migration towards the fracture center results in the increase in the depth of particle penetration into the fracture and the suppression of gravitational convection in the vicinity of the leading front. The calculations are compared with available experimental data and an analytical formula for the height of the dense packed sediment. A good agreement between the analytical theory, the experiments, and the two-dimensional calculations is attained.

Multi-fluid model of suspension filtration in a porous medium

In the framework of a three-fluid approach, a new model of suspension filtration in a porous medium is constructed with account for the formation of a dense packing of trapped particles with finite permeability and porosity. The following three continua are considered: the carrier fluid, the suspended particles, and the deposited particles. For a one-dimensional transient flow of suspension, a system of equations for the concentrations of the suspended and deposited particles, the suspension velocity, and the pressure is constructed. Two cases of the flow in a porous medium are considered: plane and radial. Numerical solution is found using a finite-difference method. Numerical calculations are shown to be in agreement with an analytical solution for the simplest case of filtration with a constant velocity and constant porosity and permeability. A comparison is performed with the classic filtration models for five sets of experimental data on the contamination of a porous sample. It is shown that near the inlet boundary, where an intense deposition of particles takes place, the new model describes the concentration profile of the deposited particles more accurately than the classical model.

Damage to formation surrounding flooding wells: modelling of suspension filtration with account of particle trapping and mobilization

We consider the filtration of raw water in a formation surrounding injection wells in oilfields of Western Siberia. The mathematical model for suspension filtration developed earlier on the basis of tree-continua approach allows to describe the permeability damage and recovery due to trapping and mobilization of externally-introduced fines. As compared to classical deep-bed filtration models, the proposed model takes into account the filtration of the carrier fluid through the pack of trapped fines and uses only two free parameters to describe the particle trapping and mobilization rates. It has gone through a thorough validation campaign against experimental data with contamination of porous samples by external fines and mobilization of pre-seeded particles in sand packs. Simulations of permeability dynamics in the zone surrounding injection wells are carried out using the values of free parameters obtained by tuning the model against available lab experiments. Both continuous and periodic water flooding/cleanup is modelled. It is found that there are periodic regimes of water injection, in which the permeability of the rock is not damaged. The study will be continued after the calibration of the model against thorough laboratory tests with natural cores and field tests of injection rate dynamics in flooding wells.

Optimal disturbances of a dusty-gas plane-channel flow with a nonuniform distribution of particles

The classical stability theory for multiphase flows, based on an analysis of one (most unstable) mode, is generalized. A method for studying an algebraic (non-modal) instability of a disperse medium, which consists in examining the energy of linear combinations of three-dimensional modes with given wave vectors, is proposed. An algebraic instability of a dusty-gas flow in a plane channel with a nonuniform particle distribution in the form of two layers arranged symmetrically with respect to the flow axis is investigated. For all possible values of governing parameters, the optimal disturbances of the disperse flow have zero wavenumber in the flow direction, which indicates their banded structure (“streaks”). The presence of dispersed particles in the flow increases the algebraic instability, since the energy of optimal disturbances in the disperse medium exceeds that for the pure-fluid flow. It is found that for a homogeneous particle distribution the increase in the energy of optimal perturbations is proportional to the square of the sum of unity and the particle mass concentration and is almost independent of particle inertia. For a non-uniform distribution of the dispersed phase, the largest increase in the initial energy of disturbances is achieved in the case when the dust layers are located in the middle between the center line of the flow and the walls.

Hydrodynamic stability of stratified suspension flow in a plane channel

Hydrodynamic stability of particle-laden flows was studied earlier within the two-fluid approach mostly under the assumptions that [1]: in the main flow, the concentration of the inclusions is uniform, the phase velocity slip is zero, and the particle volume fraction is negligibly small. In the case of suspension flows, where the particle-to-fluid substance density ratio is of order unity, the formulation described above has to be modified to take into account the finite volume fraction of the inclusions as well as non-uniform distribution of the particles in the main flow.

Effect of settling particles on the stability of a particle-laden flow in a vertical plane channel

The stability of a viscous particle-laden flow in a vertical plane channel in the presence of the gravity force is studied. The flow is described using a two-fluid “dusty-gas” model with negligibly small volume fraction of fines and two-way coupling of the phases. Two different profiles of the particle number density in the main flow are considered: homogeneous and non-homogeneous in the form of two layers symmetric about the channel axis. The novel element of the linear-stability problem formulation is a particle velocity slip in the main flow caused by the gravity-induced settling of the dispersed phase. The eigenvalue problem for a linearized system of governing equations is solved using the orthonormalization and QZ algorithms. For a uniform particle number density distribution, it is found that there exists a domain in the plane of Froude and Stokes numbers, in which the two-phase flow in a vertical channel is stable for an arbitrary Reynolds number. This stability domain corresponds to relatively sm...

Multigrid Pressure Solver for Proppant Transport Modelling in Hydraulic Fracturing Simulators

The multigrid method in the form of MGCS V-cycles with matrix-dependent prolongations is implemented for solving linear equations resulting from a standard discretization of a 2D elliptic equation (5-point “cross” stencil). Simplicity of input data allows incorporation of the solver into any relevant commercial or research code. The performance of MGCS is compared against other iterative (in-house BiCGStab preconditioned by ILU(2) factorization, PyAMG) and direct (DSS from Intel MKL library) solvers. The following performance tests are considered: 1) synthetically-generated matrices with large contrast in elements; 2) matrices formed during simulations of multiphase flows in hydraulic fractures/slots with large viscosity contrast (up to 106). Dependence of the performance of MGCS on the number of smoothing iterations at each mesh level and aspect ratio of mesh cells is studied. It is found that the best performance is gained by MGCS V-cycles with a single smoothing iteration per mesh level on meshes with square cells. Increasing the aspect ratio of the mesh cells and increasing the number of smoothing iterations per mesh level typically slows down the convergence rate. The MGCS code shows the best performance as compared to other solvers tested. The speed-up increases significantly with an increase in the mesh resolution.

Flow of viscoplastic suspensions in a hydraulic fracture: implications to overflush

The study is devoted to modeling of multiphase flows of immiscible viscoplastic fluids in a hydraulic fracture. In the framework of the lubrication approximation, three-dimensional Navier-Stokes equations are reduced to hyperbolic transport equations for the fluid tracers and a quasi-linear elliptic equation in terms of the fluid pressure. The governing equations are solved numerically using the finite-difference approach. A parametric study of the displacement of Bingham fluids in a Hele-Shaw cell is carried out. It is found that fingers developed through the pillar of a yield-stress suspension trigger the development of unyielded zones. An increase in the Bingham number leads to an increase in the so-called finger shielding effect, which manifests itself via an increase in the overall finger penetration zone and a decrease in the total number of fingers. The effect of flow parameters on the displacement of hydraulic fracturing proppant-laden suspension by a clean fluid in the vicinity of the perforation zone is carried out. This particular case is considered in application to overflush at the end of a stimulation treatment, when a small portion of a thin clean fluid is injected to wash out the particles from the wellbore into the fracture. It is found that an increase in the yield stress and the viscosity contrast between the fracturing and the overflush fluids typically reduces the area of the cavity thus mitigating the risk of loosing the conductive path between the wellbore and the fracture after the fracture closure.