Andrei A. Osiptsov

ASYMPTOTIC MODEL OF THE INERTIAL MIGRATION OF PARTICLES IN A DILUTE SUSPENSION FLOW THROUGH THE ENTRY REGION OF A CHANNEL

The inertial migration of particles in a dilute suspension flow through the entry region of a plane channel (or a circular pipe) is considered. Within the two-fluid approach, an asymptotic one-way coupling model of the dilute suspension flow in the entry region of a channel is constructed. The carrier phase is a viscous incompressible Newtonian fluid, and the dispersed phase consists of identical noncolloidal rigid spheres. In the interphase momentum exchange, we take into account the drag force, the virtual mass force, the Archimedes force, and the inertial lift force with a correction factor due to the wall effect and an arbitrary particle slip velocity. The channel Reynolds number is high and the particle-to-fluid density ratio is of order unity or significantly larger unity. The solution is constructed using the matched asymptotic expansion method. The problem of finding the far-downstream cross-channel profile of particle number concentration is reduced to solving the equations of the two-phase bound...

The inertial lift on a spherical particle settling in a horizontal viscous flow through a vertical slot

The inertial lift force exerted on a small rigid sphere settling due to gravity in a horizontal channel flow between vertical walls is investigated. The method of matched asymptotic expansions is used to obtain solutions for the disturbance flow on the length scales of the particle radius and the channel width (inner and outer regions, respectively). The channel Reynolds number is finite, while the particle Reynolds numbers that are based on the slip velocity and the mean shear rate are small. The inner flow is described by the linear Stokes equations. The outer problem is governed by the linear Oseen-like equations with the particle effect approximated by a point force. The outer equations are solved numerically using the two-dimensional Fourier transform of the disturbance velocity field. The lift coefficient is evaluated as a function of governing dimensionless parameters: the particle coordinate across the channel, the channel Reynolds number, and the slip parameter. The particle always migrates away ...

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.

Unified graph-based multi-fluid model for gas-liquid pipeline flows

We present a novel model of transient multiphase flow for gas-liquid mixtures in long pipelines. The study develops a general flexible formulation of a one-dimensional flow for complex multiphase mixtures in the presence of slip and mass exchange. Multi-fluid and drift-flux approaches are combined. The multi-fluid model governs the momentum balance for the fluids, and drift-flux correlations are used for modelling of the slip between individual components. A tree-like graph is introduced to describe the hierarchy of fluids and components within the mixture. The numerical implementation for an arbitrary number of components is based on the extension of the Semi-Implicit Method for Pressure-Linked Equations. Imbalances of equations are selected as criteria for iteration convergence. Specific corrections are discussed, including those proposed for the equilibrium gas-release model.

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.

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.