Filtration Problem with Nonlinear Filtration and Concentration Functions

Ancient architectural buildings are of great value for all modern humanity. Over time, under the influence of vibrations, water and other man-made and natural factors, the foundations of such buildings are destroyed, the soil structure changes. Currently, one of the most popular methods of strengthening soils and strengthening foundations is the jet grouting technology. When the liquid grout passes through the porous rock, the suspended particles of the grout form a deposit. In this paper, we study a one-dimensional model of suspension deep bed filtration in a porous medium with different particle capture mechanisms. The considered filtration model consists of the balance equation for the masses of suspended and retained particles and the kinetic equation for deposit growth. In this case, the deposit growth rate is determined by the concentration function of suspended particles, which, in turn, depends on the properties of the suspension and the geometry of the porous medium. The solution to the problem is obtained for linear and non-linear concentration functions. An asymptotic solution of the problem is constructed for both types of functions near the concentration front of suspended and retained particles. It is shown that the asymptotic and numerical solutions are close over a long time interval.