MATHEMATICAL MODELS OF DEEP BED FILTRATION IN A POROUS MEDIUM

Deep bed filtration of suspensions and colloids in a porous medium is considered. The mathematical macroscopic model of one-dimensional filtration in a homogeneous porous medium includes the equation of mass balance and the kinetic equation of deposit growth. Two first-order equations with unknown concentrations of suspended and retained particles form a hyperbolic system. The simplest filtration model is considered in detail, its exact solution in integral form, the monotonicity of the solutions and the asymptotics for large time. Complicated filtration model takes into account changes in the properties of the porous medium with increasing deposit. The solution of the problem in integral form is derived. The condition for the solvability of the system and the Riemann invariant connecting solutions on the characteristics are presented. If the filtration function has a root of order less than 1, the time for complete blocking of small pores is finite. The solution has a weak singularity on the blocking line, which is not a characteristic of a hyperbolic system. The new model, taking into account the concentration of vacant spaces for trapping particles on the porous medium frame, includes 3 equations with 3 unknowns. The solution to the system is given in integral form. For the filtration of a binary system with two types of particles, the dependence of the solution profiles on time is studied. It is shown that the profile monotonicity changes. Members of the Department of Applied Mathematics of the National Research University MGSU have been studying filtration models for the past few years. The latest results and unsolved problems are listed.