Grout filtration in porous soil is used in construction industry to create underground waterproof walls. When the suspension flows through the pores, various forces act on the suspended particles, blocking them on the frame of the porous medium. A one-dimensional model of deep bed filtration for a monodisperse suspension in a porous medium with several particle capture mechanisms is considered. The mathematical model includes the equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth with a piecewise-smooth linear-constant filtration function and a nonlinear concentration function. The solution of the nonlinear model is obtained by the finite difference method using an explicit difference scheme with second-order approximation. To construct the asymptotics of a complex model, the solutions of simplified linear and semilinear models and their combination are used. In the zone of a smooth filtration function, the best approximation of the solution of a complex model is determined by a certain linear combination of simple solutions. In another area, solution of a simplified problem with a piecewise-smooth filtration function and a linear concentration function is closest to the solution of a nonlinear model. Calculations show that in the zone of a smooth filtration function, a combination of simple solutions defines a solution approximation with second-order of smallness. In the area where it is necessary to take into account the non-smoothness of the filtration function, the approximation of a solution has a first order of smallness.