A numerical technique for solving an initial-boundary value problem of the single compressible fluid flow through elastic porous medium is presented. The model of compressible fluid flow in elastic porous medium is derived within the theory of hyperbolic thermodynamically compatible systems of conservation laws that provides a formulation of mathematically and physically well-posed governing equations of physical processes. Numerical method is based on the WENO-Runge-Kutta algorithm. For the boundary flux evaluation the characteristic decomposition and linearized boundary conditions are used. A two-dimensional problem of the fluid inflow into the saturated porous rectangular domain from its left boundary is studied numerically.