Calculation of the filtration problem by finite differences methods

Yuri Osipov,Yuri Galaguz,A Volkov,A Adamtsevich,A Pustovgar,Galina Safina

doi:10.1051/matecconf/201825104021

Yuri Osipov, Yuri Galaguz + Show 4 more

Open Access

https://doi.org/10.1051/matecconf/201825104021

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Abstract

The filtration problem of a suspension in a porous medium is relevant for the construction industry. In the design of hydraulic structures, construction of waterproof walls in the ground, grouting the loose soil, it is necessary to calculate the transfer and deposition of solid particles by the fluid flow. A one-dimensional filtration problem of a monodisperse suspension in a porous medium with a size-exclusion capture mechanism is considered. It is assumed that as the deposit grows, the porosity and admissible flow of particles through the porous medium change. The solution of the initial filtration model and the equivalent equations are calculated. For the numerical calculation of the problem, both standard first-order finite difference formulas and more accurate second-order schemes were used. The obtained solutions are compared with the results given by the TVD-scheme.

Highlights

The problems associated with the transport of small solid particles by the fluid flow and the deposition of particles in the pores of a porous medium are relevant for many technologies and industries
The deep bed filtration of the monodisperse suspension in a homogeneous porous medium is considered in the paper
A traditional mathematical model determining the one-dimensional filtration of an incompressible monodisperse suspension in a porous medium with a size-exclusion mechanism of particles capture relates the concentrations of suspended and retained particles to a system of two first-order partial differential equations

Summary

Introduction

The problems associated with the transport of small solid particles by the fluid flow and the deposition of particles in the pores of a porous medium are relevant for many technologies and industries. A traditional mathematical model determining the one-dimensional filtration of an incompressible monodisperse suspension in a porous medium with a size-exclusion mechanism of particles capture relates the concentrations of suspended and retained particles to a system of two first-order partial differential equations. Replacing differential equations by difference equations allows to construct an explicit difference scheme with first-order approximation The disadvantages of this scheme are a small step in time for the fulfillment of the convergence condition and the low accuracy of the numerical solution, especially near the concentrations front where the solution is discontinuous. The use of more complex counter-current schemes and rapidly converging Lax-Vendroff schemes leads to unjustified smoothing or non-physical oscillations of the solution near the line of discontinuity [14] These difference schemes, as well as TVD-schemes (Total Variation Diminishing scheme) in application to filtration problems have the first order of approximation [15]. These numerical solutions are compared with the solutions based on the standard first-order schemes and the TVD scheme

Mathematical model

Finite difference schemes of the filtration problem

Numerical calculation

Conclusion