<p>We propose a nonlinear version of the mathematical model of convective diffusion aimed at description of variations of the water hardness in the course of filtration of water in a porous body whose skeleton is formed by ion-exchange resins (kainites) or crushed zeolites or coal. The accompanying sorption-desorption processes are treated as nonlinear sources for a limited capacity of the skeleton. For the nonlinear equations of the model, we formulate the corresponding problems of mathematical physics and carry out the numerical analysis of their solutions constructed for different physical conditions. It is shown that the contribution of nonlinear parts to the obtained solutions for concentrations is insignificant in most cases. The strongest influence on the difference between the solutions of linear and nonlinear boundary-value problems is exerted by the coefficient of convective transfer rate. It is shown that the choice of the zero-order approximation in the constructed Neumann series, which are the solutions of the original nonlinear boundary-value problem, exerts practically no influence on the concentration of admixtures for small times of filtration, convective transfer rate, and skeleton capacity.</p>