The main goal is to numerically express Biot's equations for wave propagation in porous media with moment-tensor wave-excitation, to implement the numerical method, and to illustrate the loss mechanism when a fast P-wave propagates in a homogeneous medium. The numerical problem is solved with the method introduced and applied earlier for elastic and poroelastic problems. That is, the second order differential Biot equations in displacement- and stress-field variables are changed to first-order differential equations in the velocity- and stress-fields. The 2D finite difference scheme is implemented in a staggered grid in time and space with fourth order derivatives in space and second order derivatives in time. The code implementation of derivatives, accuracy, stability, convergence, initial conditions, moment tensor source, and boundary conditions are analyzed. The convergence is verified by the method of manufactured solutions. The source function is formulated as a force in time and location and added on the right hand side of Biot's equations. Two homogeneous porous media with consolidated sandstone frames and saturating pore fluid consisting of water or gas are tested. The source function chosen in the tests gives super-seismic frequencies. The water-filled pores give large amplitude slow Ps-waves in mainly wave mode. On the contrary, the gas-filled pores give slow Ps-energy in mainly diffusive mode. In conclusion, the numerical tests show expected behavior of the by Biot given poroelastic wave propagation. The real setting is a deep solid layer with a reservoir consisting of fluctuations of sand and shale. The pores are filled with water, oil, liquid CO\(_{2}\), or gas. Using well measurements a real geological model can be obtained and the effect on seismic and super-seismic data of saturating fluid in a reservoir observed from the earth surface or in a well-bore and in the laboratory (in core-plugs), respectively, can be analysed.Keywords --- poroelastic modelling, Biot, full waveform, mathematical formulation, numerical study, Rock physics, borehole geophysics