Modeling of suspension displacement process

Subject: transport of fluid containing suspended solid particles significantly affects the strength and stability of underground storage facilities, tunnels and hydraulic structures. The process of suspension filtration and displacement of suspension by a flow of fluid is considered in this article. Research background: filtration problems have been intensively studied for the last half-century. During this period, filtration models have become much more advanced. When modeling long-term deep bed filtration, modern researchers have to take into account the numerous factors that influence the transport and deposition of microscopic particles in the porous media. A number of models are being constructed on the basis of balance relationship between suspended and retained particles. Stochastic approaches to filtration problems using the Boltzmann model, network models and random walk equations are also successfully being developed. Research objectives: the study of an advanced one-dimensional model of suspension filtration in a solid porous medium when the suspension is being displaced with pure water. Materials and methods: we consider the process of displacement of suspension with pure water in a porous medium at which the transfer of fine particles and the accumulation of a deposit occur. The mechanical and geometric interaction of particles with a porous medium is the basis of our mathematical model: the solid particles freely pass through the large pores and get stuck in the pores whose size is smaller than the particle diameter. It is assumed that the fluid flow or other particles cannot knock out the retained particles. Deep bed filtration model is described by the equation of mass balance of suspended and retained particles of suspension and the kinetic equation for growth of deposit. When deep bed filtration process is long, the number of free small pores is significantly reduced, which leads to the changes in permeability and porosity of the porous medium. In order to account for this phenomenon, in contrast to the classical filtration equations, the dependence of the coefficients of mass balance equation on deposit concentration is introduced. In this problem at the initial moment a porous medium is filled with a suspension of retained and suspended particles at given concentrations. At filter inlet the pure water starts flowing, which displaces the suspension and gradually fills the porous medium. In the porous medium with pure water the filtering of suspension is terminated, the suspended particles concentration becomes zero, and the retained particles concentration is constant. The numerical calculation is performed by the method of finite differences. Results: for the deep bed filtration problem with variable porosity and permeability, a moving boundary between two phases has been identified, i.e., the front of the moving water flow, and its graph is constructed. Three-dimensional plots of retained and suspended particles concentrations and plots of their two-dimensional cross-section at a fixed time and for a prescribed distance from the filter input are created. The numerical solution is compared with the exact solution for the case of constant coefficients. Conclusions: it is shown that the filtration model with constant functions of porosity and permeability for small values of time can be a linear approximation of more general nonlinear models. Practical significance: planning and development of modern technologies for wastewater and industrial waste treatment, protection of underground structures from groundwater and flood waters, strengthening of porous soil by the concrete grouting method are based on the results of mathematical modeling of filtration problems. The results of the paper allow us to reduce the amount and cost of laboratory research and optimize the cleaning technologies of filter systems.