Numerical solution of one-dimensional direct problems for the propogation of cross waves in porous medium
In the paper, the actual problem associated with the propagation of waves in an acoustic medium is discussed. Its solution is also successfully applied to various tasks of geophysics, biomechanics, fluid filtration in porous medium, etc., where the porous medium consists of an elastically deformable matrix filled with liquid. To carry out a complex study of the propagation of seismic cross waves in a porous medium, a mathematical model is developed based on the basic rules of conservation of mass, momentum, and energy, taking into account the energy absorption due to the coefficient of inter-component friction. The adequacy of this model makes it possible to explain the observed properties of rocks in seismic studies. By introducing a new variable, instead of a second-order wave equation, we obtain a hyperbolic system of equations of the first order, which has two families of characteristics that are straight lines with slopes +1 and -1, and recursive formulas for numerical integration of the problem. For carrying out computing experiments on a computer, a software tool was developed and a trial model problem was solved. The results of numerical calculations are presented in the form of graphs