Modeling of steady-state liquid filtration in a piecewise inhomogeneous elastic-porous medium in the class of almost-periodic functions (plane problem)

When modeling liquid filtration in a porous medium, it is assumed that the filtration coefficient is constant, as a result of which the solution is simplified and reduced to a boundary value problem for the Laplace equation. In this paper, the almost periodic of Bohr analytical solutions of the plane problem of steady-state liquid filtration in an elastic - porous piecewise inhomogeneous domain are constructed using a generalized discrete Fourier transform . The domain is a strip consisting of several layers (strips) with different elastic and filtration parameters. Assuming that the filtration coefficient of an elastic-porous medium depends on the first invariant of the stress tensor, we consider it linearly dependent on the coordinate varying along the bandwidth. The filtration problem is reduced to solving a system of partial differential equations with specified boundary conditions on the upper and lower boundaries of the entire multilayer strip and conditions on the internal lines of the media interface, which in turn is reduced to solving the Cauchy problem for a system of Bessel ordinary differential equations. All solutions in this paper are obtained in the form of absolutely convergent Bohr-Fourier series, the coefficients of which are expressed in terms of given functions. Fluid filtration in a three-layer strip consisting of various light and sufficiently elastic-porous sedimentary and igneous rock layers is modeled. Graphs of the desired mechanical parameters are constructed. Their convergence to boundary conditions and conditions on the interface lines of media is shown. The paper also provides basic information concerning the properties of almost-periodic functions and the generalized discrete Fourier transform necessary for a more detailed understanding of the problem.