Seismic methods based on the seismic waves propagation in an acoustic or an ideally elastic medium, were successfully applied to various geophysical problems to identify geological structures. In such studies, properties of a pore liquid such as density, the module of volumetric deformation, fluidsaturation and viscosity were generally ignored. A porous medium of consisting it, is an elastic, i.e., deformable matrix filled with a viscous liquid, being a realistic model which allows us to explain observable effects of seismic research of properties of rocks in the presence of a pore liquid. Recently the numerical simulation of seismic wave propagation in fluidsaturated liquid porous media, has received a special attention because of its practical application in various areas of problems of geophysics, biomechanics and oil reservoir characterization. In the case, the Frenkel-Biot model is generally used as a basis. A characteristic feature of models of this type alongside with distribution of transverse and longitudinal seismic waves, is the presence of an additional second longitudinal wave. The speeds of propagation of such waves are functions of four elastic parameters in the Frenkel-Biot theory for preset values of physical density of a solid matrix, a saturating liquid and porosity. In 1989, V.N. Dorovsky based on the common first physical principles, constructed nonlinear mathematical model for porous media. Just as in the Frenkel-Biot theory, in the Dorovsky model there are three types of the sound oscillations: transverse and two types of longitudinal. As opposed to models of the Frenkel-Biot type, in the linearized Dorovsky models a medium is described by three elastic parameters. These elastic parameters in one-to-one correspondence are expressed by three speeds of seismic wave propagations. This circumstance is important for the numerical modeling of elastic waves in a porous medium when distributions of speeds of acoustic waves and physical density of the matrix of saturating liquids and porosity are known.
Finite difference methods of solving of problems for the Biot equation system have been formulated in several ways, these are: the central difference finite difference method in terms of displacement, the predictor-corrector finite difference method for the velocity-stress system of the equations.
In this paper, the system of linearized equations of porous media in the absence of dissipation of energy in 2D heterogeneous is case numerically solved. The initial system of equations as first order hyperbolic system in terms of velocity of a solid matrix, velocity of a saturating liquid, solid stress, and fluid pressure. For the numerical solution of the task in question, the method of combination of analytical Laguerre transformation and a finite difference method is used.