Justification of the asymptotic behavior of the filtration theory regimes by the method of conditional moment

In the work, a substantiation of the filtration modes is proposed by the method of conditional moments. The Navier-Stokes stationary equation of motion is used as initial equation. The calculations within the framework of this model include the construction of a solution of stochastic equations by the Green's function method, the subsequent conditional averaging of this solution, the direct and inverse Fourier transform in order to extract the integrals containing the structure of the medium and to obtain the desired averaged equations, respectively. At large Reynolds numbers, inertial terms dominate. When averaging them, a structural form of the quadratic filtration law is obtained. At low Reynolds numbers, the viscous terms make the main contribution, the averaging of which leads to Darcy's law. (paper)