Обратная задача для линейной функции фильтрации
Filtration problem for a suspension in a porous medium characterize the injection of a liquid grout into porous rock to strengthen loose soil or create waterproof partitions during the construction of tunnels and underground structures. The suspension is injected under pressure into an empty homogeneous porous medium and moves from inlet to outlet. Some particles get stuck in the pores and form a fixed deposit. The macroscopic model of filtration includes the equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth. For a one-dimensional model with a linear filtration function, an asymptotic solution is constructed near the concentration front of suspended and retained particles. A small parameter of the asymptotic expansion is a characteristic variable proportional to the distance to the concentration front. On the basis of explicit asymptotic formulas, the inverse filtration problem is solved - finding the filtration function from a given concentration of suspended particles at the outlet of a porous medium. The coefficients of the filtration function are obtained by the least squares method from the condition of the best approximation of the numerical solution by the asymptotics. It is shown that the calculated parameters are close to the coefficients of the model, and the asymptotics well approximates the numerical solution.