Particle transport in a porous medium with initial deposit
Modelling of the suspended particles transport in a porous medium is used in the analysis of methods for strengthening foundations. To strengthen the porous soil, a low concentration cement-based grout is pumped into it under pressure. The suspension is filtered in a porous medium and fills the cavity of the soil. Grains of the grout are distributed along the network structure of the porous medium and strengthen the soil. The transfer of particles by a flow of a carrier fluid is accompanied by an uneven formation of a deposit on the porous medium frame. The purpose of the paper is to determine the mobile two-phase boundary between water and the particles during the injection of a suspension into a porous medium and to obtain an analytical solution of the nonlinear filtration problem for a general case of variable porosity and permeability. The mathematical model of one-dimensional deep bed filtration with size-exclusion particles retention includes the equations of mass balance and kinetic rate of a deposit and unsteady boundary conditions with unknown dimensionless concentrations of suspended and retained particles. Methods of non-linear asymptotic analysis are used to obtain the analytical solution and to construct an asymptotics near the porous medium inlet. The asymptotics is determined on the basis of a local exact solution of the problem. It is shown that the mobile two-phase boundary moves with variable speed. At the boundary of two phases, the concentration of suspended particles is discontinuous, and the concentration of retained particles is continuous and loses its smoothness. The exact explicit formula for the two-phase boundary and its asymptotics in a form convenient for calculations are obtained. An exact solution is obtained on the boundary of the porous medium and the asymptotics of the filtration problem is constructed near the porous medium inlet. Numerical calculation of the asymptotic solution is performed; graphs of the dependence of concentrations on time and coordinate are presented. In contrast to the numerical solution, analytical methods make it possible to determine the dependence of the solution of the filtration problem on the controlled external parameters. This allows the construction engineers to choose the best size of injected grout grains and the properties of the carrier fluid, optimize the filtration process and form a grouted porous soil of the required strength and density.