The Biot model of poroelasticity is considered as a base for the algorithm of numerical estimation of the attenuation of seismic waves in fluid-saturated media. The first-order partial derivative equations system of the Biot model describes both the velocities of fluid and solid phases at every point of the continuum, as well as the total stress tensor of the undrained porous material and fluid pressure. The system is approximated with central differences on a staggered grid, where different components of the solution vector are located in different node groups. The balance method is applied to the coefficients of the equations to deal with discontinuities appearing in heterogeneous media. To avoid non-physical reflections on the boundaries, the Biot system is modified to form perfectly matched layers. Several numerical experiments are provided to demonstrate the application of the developed solver in the framework of seismic attenuation estimation in dependence on the geometrical structure and physical properties of the media.