Abstract
The problems of underground fluid mechanics play an important role in the design and preparation for the construction of tunnels and underground structures. To strengthen the insecure soil a grout solution is pumped under pressure in the porous rock. The liquid solution filters in the pores of the rock and strengthens the soil after hardening. A macroscopic model of deep bed filtration of a monodisperse suspension in a porous medium with a size-exclusion mechanism for the suspended particles capture in the absence of mobilization of retained particles is considered. The solids are transported by the carrier fluid through large pores and get stuck at the inlet of small pores. It is assumed that the accessibility factor of pores and the fractional flow of particles depend on the concentration of the retained particles, and at the initial moment the porous medium contains an unevenly distributed deposit. The latter assumption leads to inhomogeneity of the porous medium. A quasilinear hyperbolic system of two first-order equations serves as a mathematical model of the problem. The aim of the work is to obtain the asymptotic solution near the moving curvilinear boundary - the concentration front of suspended particles of the suspension. To obtain a solution to the problem, methods of nonlinear asymptotic analysis are used. The asymptotic solution is based on a small-time parameter, measured from the moment of the concentration front passage at each point of the porous medium. The terms of the asymptotics are determined explicitly from a recurrent system of ordinary differential and algebraic equations. The numerical calculation is performed by the finite difference method using an explicit TVD scheme. Calculations for three types of microscopic suspended particles show that the asymptotics is close to the solution of the problem. The time interval of applicability of the asymptotic solution is determined on the basis of numerical calculation. The constructed asymptotics, which explicitly determines the dependence on the parameters of the system, allows to plan experiments and reduce the amount of laboratory research.
Highlights
The study of suspension filtration in a porous medium is an important problem for many areas of science and technology
When injected at the porous medium inlet, some suspended particles are transported by the fluid flow through the porous sample and leave the porous medium at the outlet, others get stuck in narrow pores and form a deposit
It is shown that the solution of the filtration problem satisfies natural physical conditions: the concentration of retained particles is constant in time before the concentration front and increases in time behind the front; the suspended particles concentration is zero before the front and is positive behind the concentration front
Summary
The study of suspension filtration in a porous medium is an important problem for many areas of science and technology. The filtration process is accompanied by the formation of deposit in the pores, which changes the structure and properties of the porous medium [1]. When solid suspended particles are transported by the fluid flow, some particles are retained on the framework of the porous medium. Depending on the properties of the suspension and the porous medium, mechanical interaction, diffusion, viscosity, electrostatic and gravitational forces can play an important role in the capture of particles [3]. Deposit is formed in the entire porous medium, and in its surface layer. When injected at the porous medium inlet, some suspended particles are transported by the fluid flow through the porous sample and leave the porous medium at the outlet, others get stuck in narrow pores and form a deposit
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