Abstract
Filtration problems are actual for the design of underground structures and foundations, strengthening of loose soil and construction of watertight walls in the porous rock. A liquid grout pumped under pressure penetrates deep into the porous rock. Solid particles of the suspension retained in the pores, strengthen the loose soil and create watertight partitions. The aim of the study is to construct an explicit analytical solution of the filtration problem. A one-dimensional model of deep bed filtration of a monodisperse suspension in a homogeneous porous medium with size-exclusion mechanism of particles retention is considered. Solid particles are freely transferred by the carrier fluid through large pores and get stuck in the throats of small pores. The mathematical model of deep bed filtration includes the mass balance equation for suspended and retained particles and the kinetic equation for the deposit growth. The model describes the movement of concentrations front of suspended and retained particles in an empty porous medium. Behind the concentrations front, solid particles are transported by a carrier fluid, accompanied by the formation of a deposit. The complex model has no explicit exact solution. To construct the asymptotic solution in explicit form, methods of nonlinear asymptotic analysis are used. The new coordinate transformation allows to obtain a parameter that is small at all points of the porous sample at any time. In this paper, a global asymptotic solution of the filtration problem is constructed using a new small parameter. Numerical calculations are performed for a nonlinear filtration coefficient found experimentally. Calculations confirm the closeness of the asymptotics to the solution in the entire filtration domain. For a nonlinear filtration coefficient, the asymptotics is closer to the numerical solution than the exact solution of the problem with a linear coefficient. The analytical solution obtained in the paper can be used to analyze solutions of problems of underground fluid mechanics and fine-tune laboratory experiments.
Highlights
The problems associated with the transport and deposition of solid microscopic particles in porous media are actual in nature, science and technology
The paper considers a one-dimensional model of filtration of a monodisperse suspension in a homogeneous porous medium
The model of deep bed filtration is determined by a nonlinear hyperbolic system of firstorder equations, which does not have an exact solution in explicit form
Summary
The problems associated with the transport and deposition of solid microscopic particles in porous media are actual in nature, science and technology. Suspended solid particles are transported by the carrier fluid through the pores, with some particles blocking and forming a deposit. Depending on the structure and properties of the porous medium and suspended particles, diffusion, electrostatic, gravitational and hydrodynamic forces, etc., affect the particles retention. It is assumed that the retained particles cannot be knocked out of the pore by other particles or by the flow of a carrier fluid [7]
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