Nonlinear wave dynamics of an elastically deformed saturated porous media is investigated following the Blot approach. Mathematical models under research are the Blot model and its generalization by consideration of viscous stresses inside liquids. Using two-scales and linear WKB methods, the classical Blot system is transformed to a first-order wave equation. To construct the solution of the other system, an asymptotic modified two-scales method is developed. Initial system of equations is transformed to a nonlinear generalized Korteweg-de Vries-Burgers equation for quick elastic wave. Distinctions of wave propagation in the context of the Blot model and its generalization are shown.