A computational model for the small amplitude wave propagation in an elastic porous medium saturated by the viscous compressible fluid is discussed. The presented model is an extension of the model [1] and its derivation is based on the symmetric hyperbolic thermodynamically compatible system for two-phase solid-fluid mixture with finite deformations of the solid phase. In the present consideration, the fluid viscosity is taken into account via the unified hyperbolic model for viscous flows with shear stress relaxation. The governing equations form a hyperbolic system written in terms of the mixture velocity, relative velocity of phase motion, pressure and shear stress of the mixture that allows to apply an efficient finite difference method for numerical solution. Some numerical examples are presented, showing physically correct results, and, in particular, the frequency dependence of the shear wave velocity.