Abstract
The filtration problem is one of the most relevant in the design of retaining hydraulic structures, water supply channels, drainage systems, in the drainage of the soil foundation, etc. Construction of transport tunnels and underground structures requires careful study of the soil properties and special work to prevent dangerous geological processes. The model of particle transport in the porous rock, which is based on the mechanical-geometric interaction of particles with a porous medium, is considered in the paper. The suspension particles pass freely through large pores and get stuck in small pores. The deposit concentration increases, the porosity and the permissible flow of particles through large pores changes. The model of one-dimensional filtration of a monodisperse suspension in a porous medium with variable porosity and fractional flow through accessible pores is determined by the quasi-linear equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth. This complex system of differential equations has no explicit analytical solution. An equivalent differential equation is used in the paper. The solution of this equation by the characteristics method yields a system of integral equations. Integration of the resulting equations leads to a cumbersome system of transcendental equations, which has no explicit solution. The system is solved numerically at the nodes of a rectangular grid. All calculations are performed for non-linear filtration coefficients obtained experimentally. It is shown that the solution of the transcendental system of equations and the numerical solution of the original hyperbolic system of partial differential equations by the finite difference method are very close. The obtained solution can be used to analyze the results of laboratory research and to optimize the grout composition pumped into the porous soil.
Highlights
The study of water filtration in soils and various porous materials is of great practical interest in solving a variety of engineering problems in the field of hydraulic engineering, construction of water supply and wastewater facilities
Calculations of filtration problems are used in the design of retaining hydraulic structures, water supply channels, drainage systems, when draining the soil foundation, etc
Solid particles are carried by the fluid flow through the sinuous channels of a porous medium, some particles are locked in the pores and are retained on the porous frame, which leads to a change in its structure, reducing its porosity and accessible fractional flow
Summary
The study of water filtration in soils and various porous materials is of great practical interest in solving a variety of engineering problems in the field of hydraulic engineering, construction of water supply and wastewater facilities. Solid particles are carried by the fluid flow through the sinuous channels of a porous medium, some particles are locked in the pores and are retained on the porous frame, which leads to a change in its structure, reducing its porosity and accessible fractional flow. A mathematical model of particle motion in a filter based on size-exclusion mexanism of particle retention in a porous medium is considered in [5], [6]. The suspension of constant concentration is injected at the filter inlet. The concentration front of suspended and retained particles moves with a constant velocity from the inlet to the outlet. The porous medium is empty and the concentrations of suspended and retained particles are zero. The solution of the transcendental system is compared with the numerical solutions of the original hyperbolic system and the equivalent partial differential equation, obtained using difference schemes with second-order approximation
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